{"id":26,"date":"2019-01-24T18:44:50","date_gmt":"2019-01-24T18:44:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&#038;p=26"},"modified":"2025-05-27T18:36:09","modified_gmt":"2025-05-27T18:36:09","slug":"functions-and-function-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/functions-and-function-notation\/","title":{"raw":"1.1 Functions and Function Notation","rendered":"1.1 Functions and Function Notation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Determine and be able to explain whether a relation represents a function given a table or a graph.<\/li>\r\n \t<li>Evaluate functions and solve equations involving functions.<\/li>\r\n \t<li>Determine whether a function given numerically or graphically is one-to-one, and explain your rationale.<\/li>\r\n \t<li>Graph and name the functions listed in the library of functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137431376\">A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.<\/p>\r\n\r\n<div id=\"fs-id1165133394710\" class=\"bc-section section\">\r\n<h3>Determining Whether a Relation Represents a Function<\/h3>\r\n<p id=\"fs-id1165137781542\">A relation is a set of ordered pairs. The set of the first components of each <span class=\"no-emphasis\">ordered pair<\/span> is called the <strong>domain <\/strong>and the set of the second components of each ordered pair is called the <strong>range<\/strong>. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.<\/p>\r\n<p id=\"fs-id1165137676332\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{\\left(1,\\text{ }2\\right),\\text{ }\\left(2,\\text{ }4\\right),\\text{ }\\left(3,\\text{ }6\\right),\\text{ }\\left(4,\\text{ }8\\right),\\text{ }\\left(5,\\text{ }10\\right)\\right\\}[\/latex]<\/p>\r\n<p class=\"unnumbered\" style=\"text-align: left;\">The domain is [latex]\\left\\{1,\\text{ }2,\\text{ }3,\\text{ }4,\\text{ }5\\right\\}.[\/latex] The range is [latex]\\left\\{2,\\text{ }4,\\text{ }6,\\text{ }8,\\text{ }10\\right\\}.[\/latex]<\/p>\r\n<p id=\"fs-id1165134234609\">Note that each value in the domain is also known as an <strong>input<\/strong> value, or independent variable, and is often labeled with the lowercase letter [latex]x.[\/latex] Each value in the range is also known as an <strong>output<\/strong> value, or dependent variable, and is often labeled lowercase letter [latex]y.[\/latex]<\/p>\r\n<p id=\"fs-id1165137748300\">A function [latex]f[\/latex] is a relation that assigns a single value in the range to each value in the domain<em>.<\/em> In other words, no [latex]x[\/latex]-values are repeated. For our example that relates the first five <span class=\"no-emphasis\">natural numbers<\/span> to numbers double their values, this relation is a function because each element in the domain, [latex]\\left\\{1,\\text{ }2,\\text{ }3,\\text{ }4,\\text{ }5\\right\\},[\/latex] is paired with exactly one element in the range, [latex]\\left\\{2,\\text{ }4,\\text{ }6,\\text{ }8,\\text{ }10\\right\\}.[\/latex]<\/p>\r\n<p id=\"fs-id1165135421564\">Now let\u2019s consider the set of ordered pairs that relates the terms \u201ceven\u201d and \u201codd\u201d to the first five natural numbers. It would appear as<\/p>\r\n<p id=\"fs-id1165133192963\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{\\left(\\text{odd},\\text{ }1\\right),\\text{ }\\left(\\text{even},\\text{ }2\\right),\\text{ }\\left(\\text{odd},\\text{ }3\\right),\\text{ }\\left(\\text{even},\\text{ }4\\right),\\text{ }\\left(\\text{odd},\\text{ }5\\right)\\right\\}[\/latex]<\/p>\r\n<p id=\"fs-id1165135419796\">Notice that each element in the domain, [latex]\\left\\{\\text{even,}\\text{ }\\text{odd}\\right\\}[\/latex] is <em>not<\/em> paired with exactly one element in the range, [latex]\\left\\{1,\\text{ }2,\\text{ }3,\\text{ }4,\\text{ }5\\right\\}.[\/latex] For example, the term \u201codd\u201d corresponds to three values from the range, [latex]\\left\\{1,\\text{ }3,\\text{ }5\\right\\}[\/latex] and the term \u201ceven\u201d corresponds to two values from the range, [latex]\\left\\{2,\\text{ }4\\right\\}.[\/latex] This violates the definition of a function, so this relation is not a function.<\/p>\r\n<p id=\"fs-id1165135176295\"><a class=\"autogenerated-content\" href=\"#Figure_01_01_001\">Figure 1<\/a> compares relations that are functions and not functions.<\/p>\r\n\r\n[caption id=\"attachment_3023\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3023\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14135913\/d47bde493af93fa714bfc3dbe2a30008d2b7a5f01.jpeg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/> <strong>Figure 1:<\/strong> (a) This relationship is a function because each input is associated with a single output. Note that input [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n.[\/latex] (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input [latex]q[\/latex] is associated with two different outputs.[\/caption]\r\n<div id=\"fs-id1165137533627\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165135173375\">A <em><strong>function<\/strong><\/em> is a relation in which each possible input value leads to exactly one output value. We say \u201cthe output is a function of the input.\u201d<\/p>\r\n<p id=\"fs-id1165137661589\">The<strong> input values<\/strong> make up the <strong>domain<\/strong>, and the <strong>output values<\/strong> make up the <strong>range<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137445319\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137635406\"><strong>Given a relationship between two quantities, determine whether the relationship is a function.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134065124\" type=\"1\">\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify the output values.<\/li>\r\n \t<li>If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_01_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137414052\">\r\n<div id=\"fs-id1165137559269\">\r\n<h3>Example 1:\u00a0 Determining If Menu Price Lists Are Functions<\/h3>\r\n<p id=\"fs-id1165137436464\">The coffee shop menu, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_004\">Figure 2<\/a> consists of items and their prices.<\/p>\r\n\r\n<ol id=\"fs-id1165137646341\" type=\"a\">\r\n \t<li>Is price a function of the item?<\/li>\r\n \t<li>Is the item a function of the price?<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_3024\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142007\/628598f20aa79562d901ad8842b1a402adfeb5c2.jpeg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is 1.49 and a jelly donut and chocolate donut are 1.99.\" width=\"487\" height=\"233\" \/> <strong>Figure 2<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_004\" class=\"small\">\r\n<div class=\"mceTemp\"><\/div>\r\n<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135419802\">[reveal-answer q=\"fs-id1165135419802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135419802\"]\r\n<ol id=\"fs-id1165137643241\" type=\"a\">\r\n \t<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices. See <a class=\"autogenerated-content\" href=\"#Figure_01_01_027\">Figure 3<\/a>.\r\n\r\n[caption id=\"attachment_3025\" align=\"aligncenter\" width=\"731\"]<img class=\"size-full wp-image-3025\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142202\/472edeec5bc2bfb61746e8c5b3eedf15d73f1c02.jpeg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is 1.49 and a jelly donut and chocolate donut are 1.99.\" width=\"731\" height=\"241\" \/> <strong>Figure 3<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_027\" class=\"medium\">\r\n<div class=\"mceTemp\"><\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135532324\">Each item on the menu has only one price, so the price is a function of the item.<\/p>\r\n<\/li>\r\n \t<li>Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See <a class=\"autogenerated-content\" href=\"#Figure_01_01_028\">Figure 4<\/a>.\r\n<p id=\"fs-id1165137754835\">Therefore, the item is a not a function of price.<\/p>\r\n<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_3026\" align=\"aligncenter\" width=\"731\"]<img class=\"wp-image-3026 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142259\/3adba7b02c540ad82f7d8fa53691f3d781923d0f.jpeg\" alt=\"Association of the prices to the donuts.\" width=\"731\" height=\"241\" \/> <strong>Figure 4<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_01_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137437773\">\r\n<div id=\"fs-id1165135620873\">\r\n<h3>Example 2:\u00a0 Determining If Class Grade Rules Are Functions<\/h3>\r\n<p id=\"fs-id1165137442099\">In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? <a class=\"autogenerated-content\" href=\"#Table_01_01_01\">Table 1<\/a> shows a possible rule for assigning grade points.<\/p>\r\n\r\n<table id=\"Table_01_01_01\" class=\"lines\" style=\"height: 24px;\" border=\"y\" summary=\"Title of the table is \u201cClass Grades\u201d. It contains two columns and ten rows. The first column is labeled, \u201cPercent Grade\u201d, and the second column is labeled, \u201cGrade point average\u201d. Reading the rows as ordered pairs, we have: (92-100, 4.0), (87-91, 3.5), (78-86, 3.0), (72-77, 2.5), (67-71, 2.0), (62-66, 1.5), (57-61, 1.0), and (0-56, 0.0).\"><caption><strong>Table 1<\/strong><\/caption>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 114px; height: 12px;\"><strong>Percent grade<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 28px; height: 12px; text-align: center;\">0\u201356<\/td>\r\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">57\u201361<\/td>\r\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">62\u201366<\/td>\r\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">67\u201371<\/td>\r\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">72\u201377<\/td>\r\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">78\u201386<\/td>\r\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">87\u201391<\/td>\r\n<td class=\"border\" style=\"width: 42px; text-align: center; height: 12px;\">92\u2013100<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 114px; height: 12px;\"><strong>Grade point average<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 28px; text-align: center; height: 12px;\">0.0<\/td>\r\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">1.0<\/td>\r\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">1.5<\/td>\r\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">2.0<\/td>\r\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">2.5<\/td>\r\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">3.0<\/td>\r\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">3.5<\/td>\r\n<td class=\"border\" style=\"width: 42px; text-align: center; height: 12px;\">4.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1165135424616\">[reveal-answer q=\"fs-id1165135424616\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135424616\"]\r\n<p id=\"fs-id1165135260743\">For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.<\/p>\r\n<p id=\"fs-id1165137807321\">In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137588587\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_01_01_01\">\r\n<div id=\"fs-id1165135667843\">\r\n<p id=\"fs-id1165137627634\"><a class=\"autogenerated-content\" href=\"#Table_01_01_02\">Table 2<\/a>[footnote]<a href=\"http:\/\/www.baseball-almanac.com\/legendary\/lisn100.shtml\">http:\/\/www.baseball-almanac.com\/legendary\/lisn100.shtml<\/a>. Accessed 3\/24\/2014[\/footnote]\u00a0lists the five greatest baseball players of all time in order of rank.<\/p>\r\n\r\n<table id=\"Table_01_01_02\" class=\"lines\" style=\"width: 200px; height: 84px;\" border=\"y\" summary=\"Six rows and two columns. The first column is labeled, \u201cplayer name\u201d, and the second column is labeled, \u201crank\u201d. Reading the rows as ordered pairs, we have: (Babe Ruth, 1), (Willie Mays, 2), (Ty Cobb, 3), (Walter Johnson, 4), and (Hank Aaron, 5).\" cellpadding=\"0\"><caption><strong>Table 2<\/strong><\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\">Player<\/td>\r\n<td class=\"border\">Rank<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\">Babe Ruth<\/td>\r\n<td class=\"border\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\">Willie Mays<\/td>\r\n<td class=\"border\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\">Ty Cobb<\/td>\r\n<td class=\"border\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\">Walter Johnson<\/td>\r\n<td class=\"border\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\">Hank Aaron<\/td>\r\n<td class=\"border\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol id=\"fs-id1165137501241\" type=\"a\">\r\n \t<li>Is the rank a function of the player name?<\/li>\r\n \t<li>Is the player name a function of the rank?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165137724415\">[reveal-answer q=\"fs-id1165137724415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137724415\"]\r\n<p id=\"fs-id1165137682011\">a. Yes.\u00a0 The input would be the player name, and the output would be the rank.\u00a0 Each player is mapped to exactly one rank.\u00a0 This meets the definition of function.<\/p>\r\nb. Yes.\u00a0 The input would be the rank, and the output would be the player name.\u00a0 Each rank is mapped to exactly one name, so this meets the definition of function.\u00a0 However, if two players had been tied for, say, 4th place, then the name would not have been a function of rank.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134474160\" class=\"bc-section section\">\r\n<h4>Using Function Notation<\/h4>\r\n<p id=\"fs-id1165133359348\">Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <span class=\"no-emphasis\">function notation<\/span> is one representation that facilitates working with functions.<\/p>\r\n<p id=\"fs-id1165137453971\">To represent \u201cheight is a function of age,\u201d we start by identifying the descriptive variables [latex]h[\/latex] for height and [latex]a[\/latex] for age. The letters [latex]f,\\text{ }g,[\/latex] and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,\\text{ }y,[\/latex] and [latex]z[\/latex] to represent numbers and [latex]A,\\text{ }B,[\/latex] and [latex]C[\/latex] to represent sets.<\/p>\r\n<p id=\"fs-id1165135332760\" class=\"unnumbered\" style=\"padding-left: 30px;\">[latex]\\begin{array}{lllll}h\\text{ is }f\\text{ of }a\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We name the function }f;\\text{ height is a function of age}.\\hfill \\\\ h=f\\left(a\\right)\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We use parentheses to indicate the function input}\\text{. }\\hfill \\\\ f\\left(a\\right)\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We name the function }f;\\text{ the expression is read as \u201c}f\\text{ of }a\\text{.\u201d}\\hfill \\end{array}[\/latex]<\/p>\r\n<p id=\"fs-id1165137766965\">Remember, we can use any letter to name the function; the notation [latex]f\\left(a\\right)[\/latex] shows us that height, [latex]h,[\/latex] depends on age, [latex]a.[\/latex] The value [latex]a[\/latex] must be put into the function [latex]f[\/latex] to get the height. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\r\n<p id=\"fs-id1165135436660\">We can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means \u201cfirst add <em>a<\/em> and <em>b<\/em>, and the result is the input for the function <em>f<\/em>.\u201d The operations must be performed in this order to obtain the correct result.<\/p>\r\n\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<em><strong>function notation<\/strong><\/em>:\u00a0\u00a0 The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f.[\/latex] This is read as [latex]\u201cy[\/latex] is a function of [latex]x.\u201d[\/latex] The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y\\text{, }[\/latex] or [latex]f\\left(x\\right),[\/latex] represents the output value, or dependent variable.\r\n\r\n<\/div>\r\n<div id=\"Example_01_01_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135612059\">\r\n<div id=\"fs-id1165135705803\">\r\n<h3>Example 3:\u00a0 Using Function Notation for Days in a Month<\/h3>\r\n<p id=\"fs-id1165137757351\">Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137405547\">[reveal-answer q=\"fs-id1165137405547\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137405547\"]\r\n<p id=\"fs-id1165137657617\">The number of days in a month is a function of the name of the month, so if we name the function [latex]f,[\/latex] we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex] or [latex]d=f\\left(m\\right).[\/latex] The name of the month is the input to a \u201crule\u201d that associates a specific number (the output) with each input.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3027\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3027\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142437\/3934b41db8099eaa597af21c2abc98e820d55d15.jpeg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" \/> <strong>Figure 5<\/strong>[\/caption]\r\n\r\n<div id=\"Image_01_01_005\" class=\"unnumbered\"><\/div>\r\n<p id=\"fs-id1165135417826\">For example, [latex]f\\left(\\text{March}\\right)=31,[\/latex] because March has 31 days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nNote that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers or algebraic expressions as inputs and outputs.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_01_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137441910\">\r\n<div id=\"fs-id1165137527239\">\r\n<h3>Example 4:\u00a0 Interpreting Function Notation<\/h3>\r\n<p id=\"fs-id1165137526811\">A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N,[\/latex] in a town in year [latex]y.[\/latex] What does [latex]f\\left(2005\\right)=300[\/latex] represent?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137834021\">[reveal-answer q=\"fs-id1165137834021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137834021\"]\r\n<p id=\"fs-id1165137424675\">When we read [latex]f\\left(2005\\right)=300,[\/latex] we see that the input year is 2005. The value for the output, the number of police officers, [latex]N,[\/latex] is 300. Remember, [latex]N=f\\left(y\\right).[\/latex] The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were 300 police officers in the town.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134257606\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"fs-id1165137564344\">\r\n<div id=\"fs-id1165137564345\">\r\n<p id=\"fs-id1165137596424\">Use function notation to express the weight of a pig in pounds as a function of its age in days [latex]d\\text{.}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137871618\">[reveal-answer q=\"fs-id1165137871618\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137871618\"]\r\n<p id=\"fs-id1165137619935\">[latex]w=f\\left(d\\right)[\/latex] since\u00a0 the input would be days and the output would be weight.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137740780\" class=\"precalculus qa key-takeaways\">\r\n<h3>Q&amp;A<\/h3>\r\n<p id=\"eip-id1165132005171\"><strong>Instead of a notation such as [latex]y=f\\left(x\\right),[\/latex] could we use the same symbol for the output as for the function, such as [latex]y=y\\left(x\\right),[\/latex] meaning \u201c<em>y<\/em> is a function of <em>x<\/em>?\u201d<\/strong><\/p>\r\n<p id=\"fs-id1165137605080\"><em>Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as [latex]f,[\/latex] which is a rule or procedure, and the output [latex]y[\/latex] we get by applying [latex]f[\/latex] to a particular input [latex]x.[\/latex] This is why we usually use notation such as [latex]y=f\\left(x\\right),P=W\\left(d\\right),[\/latex] and so on.<\/em><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137804204\" class=\"bc-section section\">\r\n<h4>Representing Functions Using Tables<\/h4>\r\n<p id=\"fs-id1165137648317\">A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.<\/p>\r\n<p id=\"fs-id1165137761188\"><a class=\"autogenerated-content\" href=\"#Table_01_01_03\">Table 3<\/a> lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]f[\/latex] where [latex]D=f\\left(m\\right)[\/latex] identifies months by an integer rather than by name.<\/p>\r\n\r\n<table id=\"Table_01_01_03\" class=\"lines\" border=\"y\" summary=\"Two rows and thirteen columns. The first row is labeled, \u201c(input) Month number, m\u201d and the second row is labeled, \u201c(output) Days in months, D\u201d. Reading the columns as ordered pairs, we have: (1, 31), (2, 28), (3, 31), (4, 30), (5, 31), (6, 30), (7, 31), (8, 31), (9, 30) , (10, 31), (11, 30), and (12, 31).\"><caption><strong>Table 3<\/strong><\/caption>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"height: 14px; width: 129.017px;\"><strong>Month number, [latex]m[\/latex](input)<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">8<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">9<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">11<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 42.45px; text-align: center;\">12<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"height: 14px; width: 129.017px;\"><strong>Days in month, [latex]D[\/latex](output)<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">31<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">28<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">31<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">31<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">31<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">31<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">31<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 42.45px; text-align: center;\">31<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135191568\"><a class=\"autogenerated-content\" href=\"#Table_01_01_04\">Table 4<\/a> defines a function [latex]Q=g\\left(n\\right).[\/latex] Remember, this notation tells us that [latex]g[\/latex] is the name of the function that takes the input [latex]n[\/latex] and gives the output [latex]Q\\text{ .}[\/latex]<\/p>\r\n\r\n<table id=\"Table_01_01_04\" style=\"height: 51px; width: 335px;\" border=\"y\" summary=\"Two rows and six columns. The first row is labeled, \u201cn\u201d and the second row is labeled, \u201cQ\u201d. Reading the columns as ordered pairs, we have: (1, 8), (2, 6), (3, 7), (4, 6) , and (5, 8).\"><caption><strong>Table 4<\/strong><\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"height: 14px; width: 28.183px;\">[latex]n[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2167px; text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 23.2px; text-align: center;\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"height: 14px; width: 28.183px;\">[latex]Q[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">8<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 28.2167px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 23.2px; text-align: center;\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137561574\"><a class=\"autogenerated-content\" href=\"#Table_01_01_05\">Table 5<\/a> displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.<\/p>\r\n\r\n<table id=\"Table_01_01_05\" style=\"width: 523px;\" border=\"y\" summary=\"Two rows and eight columns. The first row is labeled, \u201c(input) a, age in years\u201d and the second row is labeled, \u201c(output) h, height in inches\u201d. Reading the columns as ordered pairs, we have: (5, 40), (5, 42) , (6, 44), (7, 47), (8, 50), (9, 52), and (10, 54).\"><caption><strong>Table 5<\/strong><\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>Age in years, [latex]a[\/latex] (input)<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">9<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>Height in inches,[latex]h[\/latex] (output)<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">40<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">42<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">44<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">47<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">50<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">52<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">54<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div id=\"fs-id1165137804163\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165134200185\"><strong>Given a table of input and output values, determine whether the table represents a function. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137461155\" type=\"1\">\r\n \t<li>Identify the input and output values.<\/li>\r\n \t<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_01_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137416794\">\r\n<div id=\"fs-id1165135591087\">\r\n<h3>Example 5:\u00a0 Identifying Tables that Represent Functions<\/h3>\r\n<p id=\"fs-id1165135503697\">Which table, <a class=\"autogenerated-content\" href=\"#Table_01_01_06\">Table 6<\/a>, <a class=\"autogenerated-content\" href=\"#Table_01_01_07\">Table 7<\/a>, or <a class=\"autogenerated-content\" href=\"#Table_01_01_08\">Table 8<\/a>, represents a function (if any)?<\/p>\r\n\r\n<table id=\"Table_01_01_06\" style=\"width: 150px; height: 96px;\" border=\"y\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (2, 1), (5, 3), and (8, 6).\" width=\"150\"><caption><strong>Table 6<\/strong><\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">Input<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">Output<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">2<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">5<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">8<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"Table_01_01_07\" style=\"width: 150px; height: 96px;\" border=\"y\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (-3, 5), (0, 1), and (4, 5).\"><caption><strong>Table 7<\/strong><\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<td class=\"border\">Input<\/td>\r\n<td class=\"border\">Output<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">\u20133<\/td>\r\n<td class=\"border\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">0<\/td>\r\n<td class=\"border\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">4<\/td>\r\n<td class=\"border\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"Table_01_01_08\" style=\"width: 150px; height: 96px;\" border=\"y\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (1, 0), (5, 2), and (5, 4).\"><caption><strong>Table 8<\/strong><\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th class=\"border\">Input<\/th>\r\n<th class=\"border\">Output<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">1<\/td>\r\n<td class=\"border\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">5<\/td>\r\n<td class=\"border\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">5<\/td>\r\n<td class=\"border\">4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137665675\">[reveal-answer q=\"fs-id1165137665675\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137665675\"]\r\n<p id=\"fs-id1165137401396\"><a class=\"autogenerated-content\" href=\"#Table_01_01_06\">Table 6<\/a> and <a class=\"autogenerated-content\" href=\"#Table_01_01_07\">Table 7<\/a> define functions. In both, each input value corresponds to exactly one output value. <a class=\"autogenerated-content\" href=\"#Table_01_01_08\">Table 8<\/a> does not define a function because the input value of 5 corresponds to two different output values.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWhen a table represents a function, corresponding input and output values can also be specified using function notation.\r\n\r\nThe function represented by <a class=\"autogenerated-content\" href=\"#Table_01_01_06\">Table 6<\/a> can be represented by writing\r\n<p id=\"fs-id1165137404863\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(2\\right)=1,\\text{ }f\\left(5\\right)=3,[\/latex] and [latex]f\\left(8\\right)=6.[\/latex]<\/p>\r\n<p id=\"fs-id1165137619677\">Similarly, the statements<\/p>\r\n<p id=\"fs-id1165137589116\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(-3\\right)=5,\\text{ }g\\left(0\\right)=1,[\/latex] and [latex]g\\left(4\\right)=5[\/latex]<\/p>\r\n<p id=\"fs-id1165137715365\">represent the function in <a class=\"autogenerated-content\" href=\"#Table_01_01_07\">Table 7<\/a>.<\/p>\r\n<p id=\"fs-id1165137656795\"><a class=\"autogenerated-content\" href=\"#Table_01_01_08\">Table 8<\/a> cannot be expressed in a similar way because it does not represent a function.<\/p>\r\n\r\n<div id=\"fs-id1165137749258\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"ti_01_01_03\">\r\n<div id=\"fs-id1165137698328\">\r\n<p id=\"fs-id1165137698329\">Does <a class=\"autogenerated-content\" href=\"#Table_01_01_09\">Table 9<\/a> represent a function?<\/p>\r\n\r\n<table id=\"Table_01_01_09\" style=\"height: 96px; width: 276px;\" border=\"y\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (1, 10), (2, 100), and (3, 1000).\" width=\"150\"><caption><strong>Table 9<\/strong><\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"text-align: center;\">Input<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">Output<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">100<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">1000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135322022\">[reveal-answer q=\"fs-id1165135322022\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135322022\"]\r\n<p id=\"fs-id1165137844279\">Yes.\u00a0 Each input corresponds to exactly one output.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137503241\" class=\"bc-section section\">\r\n<h3>Finding Input and Output Values of a Function<\/h3>\r\n<p id=\"fs-id1165137470651\">When we know an input value and want to determine the corresponding output value for a function, we <strong><em>evaluate<\/em><\/strong> the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.<\/p>\r\n<p id=\"fs-id1165137735634\">When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function\u2019s formula and <strong><em>solve<\/em><\/strong> for the input. Solving can produce more than one solution because different input values can produce the same output value.<\/p>\r\n\r\n<div id=\"fs-id1165137425943\" class=\"bc-section section\">\r\n<h4>Evaluation of Functions in Algebraic Forms<\/h4>\r\n<p id=\"fs-id1165137655584\">When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\\left(x\\right)=5-3{x}^{2}[\/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.<\/p>\r\n\r\n<div id=\"fs-id1165135613610\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137767182\"><strong>Given the formula for a function, evaluate. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137629040\" type=\"1\">\r\n \t<li>Replace the input variable in the formula with the value provided.<\/li>\r\n \t<li>Calculate the result.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_01_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137742220\">\r\n<div id=\"fs-id1165137455592\">\r\n<h3>Example 6:\u00a0 Evaluating Functions at Specific Values<\/h3>\r\n<p id=\"fs-id1165134193005\">Evaluate [latex]f\\left(x\\right)={x}^{2}+3x-4[\/latex] at<\/p>\r\n\r\n<ol id=\"fs-id1165137648008\" type=\"a\">\r\n \t<li>[latex]2[\/latex]<\/li>\r\n \t<li>[latex]a[\/latex]<\/li>\r\n \t<li>[latex]a+h[\/latex]<\/li>\r\n \t<li>[latex]\\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165135397244\">[reveal-answer q=\"fs-id1165135397244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135397244\"]\r\n<p id=\"fs-id1165137936905\">Replace the [latex]x[\/latex] in the function with each specified value.<\/p>\r\n\r\n<ol id=\"fs-id1165137778273\" type=\"a\">\r\n \t<li>Because the input value is a number, 2, we can use simple algebra to simplify.\r\n<p id=\"fs-id1165135160774\" class=\"unnumbered hanging-indent\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(2\\right)&amp;={2}^{2}+3\\left(2\\right)-4\\\\&amp;=4+6-4\\\\&amp;=6\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\r\n<\/li>\r\n \t<li>In this case, the input value is a letter so we cannot simplify the answer any further.\r\n<p id=\"fs-id1165137638318\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a-4[\/latex][latex]\\\\[\/latex]<\/p>\r\n<\/li>\r\n \t<li>With an input value of[latex]\\text{ }a+h,[\/latex] we must use the distributive property.\r\n<p id=\"fs-id1165137911654\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(a+h\\right)&amp;={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4\\\\&amp;={a}^{2}+2ah+{h}^{2}+3a+3h-4\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: center;\">In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\r\n<p id=\"fs-id1165135154122\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h-4[\/latex]<\/p>\r\n<p id=\"fs-id1165135632109\">and we know that<\/p>\r\n<p id=\"fs-id1165137471110\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a-4.[\/latex]<\/p>\r\n<p id=\"fs-id1165137767461\">Now we combine the results and simplify.<\/p>\r\n<p id=\"fs-id1165137573884\" class=\"unnumbered\" style=\"text-align: left;\">[latex]\\frac{f(a+h)-f(a)}{h}\\\\[\/latex]<\/p>\r\n[latex]\\begin{align*}\\text{ }\\text{ }\\text{ }\\text{ }&amp;=\\frac{({{a}^{2}}+2ah+{{h}^{2}}+3a+3h-4)-({{a}^{2}}+3a-4)}{h}&amp;&amp;\\text{ }\\\\&amp;=\\frac{2ah+{{h}^{2}}+3h}{h}\\\\&amp;=\\frac{h(2a+h+3)}{h}&amp;&amp; \\text{Factor out h.} \\\\&amp;=2a+h+3&amp;&amp;\\text{Simplify.} \\\\ \\end{align*}[\/latex]\r\n[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn Example 6, you worked with the expression below:\r\n<p style=\"text-align: center;\">[latex]\\begin{align*}\\frac{f(a+h)-f(a)}{h}\\\\\\end{align*}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">This is called the difference quotient. In Calculus, we use the difference quotient to develop an important concept called the derivative. You should work to become very comfortable with the difference quotient. We will work with it more in later sections of this chapter.<\/p>\r\n\r\n<div id=\"Example_01_01_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134043756\">\r\n<div id=\"fs-id1165137705537\">\r\n<h3>Example 7:\u00a0 Evaluating Functions<\/h3>\r\n<p id=\"fs-id1165137731385\">Given the function [latex]h\\left(p\\right)={p}^{2}+2p,[\/latex] evaluate [latex]h\\left(4\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137433651\">[reveal-answer q=\"fs-id1165137433651\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137433651\"]\r\n<p id=\"fs-id1165137433653\">To evaluate [latex]h\\left(4\\right),[\/latex] we substitute the value 4 for the input variable [latex]p[\/latex] in the given function.<\/p>\r\n<p id=\"fs-id1165137444745\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}h\\left(p\\right)&amp;={p}^{2}+2p\\\\h\\left(4\\right)&amp;={\\left(4\\right)}^{2}+2\\left(4\\right)\\\\ &amp;=16+8\\\\&amp;=24\\end{align*}[\/latex]<\/p>\r\n<p id=\"fs-id1165137785006\">Therefore, for an input of 4, we have an output of 24.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137704746\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_01_01_08\">\r\n<div id=\"fs-id1165134039322\">\r\n<p id=\"fs-id1165134039323\">Given the function [latex]g\\left(m\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{m-4},[\/latex] evaluate [latex]g\\left(5\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137441862\">[reveal-answer q=\"fs-id1165137441862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137441862\"]\r\n<p id=\"fs-id1165134037488\">[latex]g\\left(5\\right)=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_01_08\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137459747\">\r\n<div id=\"fs-id1165137459749\">\r\n<h3>Example 8:\u00a0 Solving Functions<\/h3>\r\n<p id=\"fs-id1165137460826\">Given the function [latex]h\\left(p\\right)={p}^{2}+2p,[\/latex] solve for [latex]h\\left(p\\right)=3.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132971707\">[reveal-answer q=\"fs-id1165132971707\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132971707\"]\r\n<div id=\"fs-id1165135195145\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}h\\left(p\\right)&amp;=3&amp;&amp;\\text{ }\\\\{p}^{2}+2p&amp;=3&amp;&amp;\\text{Substitute the original function.}\\\\{p}^{2}+2p-3&amp;=0&amp;&amp;\\text{Subtract 3 from each side}.\\\\\\left(p+3\\text{)(}p-1\\right)&amp;=0&amp;&amp;\\text{Factor}.\\end{align*}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165137770370\">If [latex]\\left(p+3\\right)\\left(p-1\\right)=0,[\/latex] either [latex]\\left(p+3\\right)=0[\/latex] or [latex]\\left(p-1\\right)=0[\/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[\/latex] in each case.<\/p>\r\n\r\n<div id=\"fs-id1165134114001\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}\\left(p+3\\right)=0,\\hfill &amp; p=-3\\hfill \\\\ \\left(p-1\\right)=0,\\hfill &amp; p=1\\hfill \\end{array}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165134468906\">This gives us two solutions. The output [latex]h\\left(p\\right)=3[\/latex] when the input is either [latex]p=1[\/latex] or [latex]p=-3.[\/latex] We can also verify by graphing as in <a class=\"autogenerated-content\" href=\"#Figure_01_01_006\">Figure 6<\/a>. The graph verifies that [latex]h\\left(1\\right)=h\\left(-3\\right)=3[\/latex] and [latex]h\\left(4\\right)=24.[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_3028\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3028\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142655\/5f6fe9b75ac7f991d78dac5f26509ecb4d659b86.jpeg\" alt=\"Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).\" width=\"487\" height=\"459\" \/> <strong>Figure 6<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_006\" class=\"small\">\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133056923\" class=\"precalculus tryit\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"ti_01_01_09\">\r\n<div id=\"fs-id1165134170173\">\r\n<p id=\"fs-id1165134170174\">Given the function [latex]g\\left(m\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{m-4},[\/latex] solve [latex]g\\left(m\\right)=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135664055\">[reveal-answer q=\"fs-id1165135664055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135664055\"]\r\n<p id=\"fs-id1165135664056\">[latex]m=8[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137591827\" class=\"bc-section section\">\r\n<h4>Evaluating Functions Expressed in Formulas<\/h4>\r\n<p id=\"fs-id1165137598337\">Some functions are defined by mathematical rules or procedures expressed in <span class=\"no-emphasis\">equation<\/span> form. If it is possible to express the function output with a <span class=\"no-emphasis\">formula<\/span> involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[\/latex] expresses a functional relationship between [latex]n[\/latex] and [latex]p.[\/latex] We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137827882\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165132034236\"><strong>Given a function in equation form, write its algebraic formula.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134544989\" type=\"1\">\r\n \t<li>Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves <em>only<\/em> the input variable.<\/li>\r\n \t<li>Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_01_09\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137634899\">\r\n<div id=\"fs-id1165137560474\">\r\n<h3>Example 9:\u00a0 Finding an Equation of a Function<\/h3>\r\n<p id=\"fs-id1165137452465\">Express the relationship [latex]2n+6p=12[\/latex] as a function [latex]p=f\\left(n\\right),[\/latex] if possible.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137832865\">[reveal-answer q=\"fs-id1165137832865\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137832865\"]\r\n<p id=\"fs-id1165137832868\">To express the relationship in this form, we need to be able to write the relationship where [latex]p[\/latex] is a function of [latex]n,[\/latex] which means writing it as [latex]p=\\left[\\text{expression}\\text{ }\\text{involving}\\text{ }n\\right].[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165131920640\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}2n+6p&amp;=12&amp;&amp;\\text{ }\\\\6p&amp;=12-2n&amp;&amp;\\text{Subtract }2n\\text{ from both sides}.\\\\p&amp;=\\frac{12-2n}{6}&amp;&amp;\\text{Divide both sides by 6 and simplify}.\\\\p&amp;=\\frac{12}{6}-\\frac{2n}{6}&amp;&amp;\\text{ } \\\\p&amp;=2-\\frac{1}{3}n&amp;&amp;\\text{ }\\end{align*}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135513733\">Therefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as<\/p>\r\n\r\n<div id=\"fs-id1165135187787\" class=\"unnumbered\" style=\"text-align: center;\">[latex]p=f\\left(n\\right)=2-\\frac{1}{3}n.[\/latex]<\/div>\r\n<h3>Analysis<\/h3>\r\n<div>It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[\/hidden-answer]<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_01_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135378843\">\r\n<div id=\"fs-id1165135378845\">\r\n<h3>Example 10:\u00a0 Expressing the Equation of a Circle as a Function<\/h3>\r\n<p id=\"fs-id1165137758151\">Does the equation [latex]{x}^{2}+{y}^{2}=1[\/latex] represent a function with [latex]x[\/latex] as input and [latex]y[\/latex] as output? If so, express the relationship as a function [latex]y=f\\left(x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135424684\">[reveal-answer q=\"fs-id1165135424684\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135424684\"]\r\n<p id=\"fs-id1165137937536\">First we subtract [latex]{x}^{2}[\/latex] from both sides.<\/p>\r\n\r\n<div id=\"fs-id1165134054911\" class=\"unnumbered\" style=\"text-align: center;\">[latex]{y}^{2}=1-{x}^{2}[\/latex]<\/div>\r\n<p id=\"fs-id1165133437258\">We now try to solve for [latex]y[\/latex] in this equation.<\/p>\r\n\r\n<div id=\"fs-id1165137416396\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}y=\u00b1\\sqrt{1-{x}^{2}}\\hfill \\\\ \\text{ }=+\\sqrt{1-{x}^{2}}\\text{ and }-\\sqrt{1-{x}^{2}}\\hfill \\end{array}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135369156\">We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\\left(x\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137647659\" class=\"precalculus tryit\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"fs-id1165137698725\">\r\n<div id=\"fs-id1165137698726\">\r\n<p id=\"fs-id1165137698727\">If [latex]x-8{y}^{3}=0,[\/latex] express [latex]y[\/latex] as a function of [latex]x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137677194\">[reveal-answer q=\"fs-id1165137677194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137677194\"]\r\n<p id=\"fs-id1165135344102\">[latex]y=f\\left(x\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2}3]{x}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135581166\" class=\"precalculus qa key-takeaways\">\r\n<h3>Q&amp;A<\/h3>\r\n<p id=\"eip-id1165135547539\"><strong>Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?<\/strong><\/p>\r\n<em>Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y},[\/latex] if we want to express [latex]y[\/latex] as a function of [latex]x,[\/latex] there is no simple algebraic formula involving only [latex]x[\/latex] that equals [latex]y.[\/latex] However, each [latex]x[\/latex] does determine a unique value for [latex]y,[\/latex] and there are mathematical procedures by which [latex]y[\/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[\/latex] as a function of [latex]x,[\/latex] even though the formula cannot be written explicitly.<\/em>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137648450\" class=\"bc-section section\">\r\n<h4>Evaluating a Function Given in Tabular Form<\/h4>\r\n<p id=\"fs-id1165135186424\">As we saw previously, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.<\/p>\r\n<p id=\"fs-id1165135186427\">The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See <a class=\"autogenerated-content\" href=\"#Table_01_01_10\">Table 10<\/a>.[footnote]<a href=\"http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590\">http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590<\/a>. Accessed 3\/24\/2014.[\/footnote]<\/p>\r\n\r\n<table id=\"Table_01_01_10\" style=\"height: 158px;\" border=\"y\" summary=\"Six rows and two columns. The first column is labeled, \u201cpet\u201d, and the second column is labeled, \u201cmemory span in hours\u201d. Reading the rows as ordered pairs, we have: (puppy, 0.008), (adult dog, 0.083), (cat, 16), (goldfish, 2100), and (beta fish, 3600).\" width=\"200\"><caption><strong>Table 10<\/strong><\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Pet<\/td>\r\n<td class=\"border\" style=\"width: 50%; text-align: center;\">Memory span in hours<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Puppy<\/td>\r\n<td class=\"border\" style=\"width: 50%; text-align: center;\">0.008<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Adult dog<\/td>\r\n<td class=\"border\" style=\"width: 50%; text-align: center;\">0.083<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Cat<\/td>\r\n<td class=\"border\" style=\"width: 50%; text-align: center;\">16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Goldfish<\/td>\r\n<td class=\"border\" style=\"width: 50%; text-align: center;\">2160<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Beta fish<\/td>\r\n<td class=\"border\" style=\"width: 50%; text-align: center;\">3600<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137584852\">At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P.[\/latex] The <span class=\"no-emphasis\">domain<\/span> of the function is the type of pet and the range is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of \u201cgoldfish.\u201d We would write [latex]P\\left(\\text{goldfish}\\right)=2160.[\/latex] Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[\/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.<\/p>\r\n\r\n<div id=\"fs-id1165137838337\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137870786\"><strong>Given a function represented by a table, identify specific output and input values. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137870791\" type=\"1\">\r\n \t<li>Find the given input in the row (or column) of input values.<\/li>\r\n \t<li>Identify the corresponding output value paired with that input value.<\/li>\r\n \t<li>Find the given output values in the row (or column) of output values, noting every time that output value appears.<\/li>\r\n \t<li>Identify the input value(s) corresponding to the given output value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_01_11\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137619419\">\r\n<div id=\"fs-id1165137619421\">\r\n<h3>Example 11:\u00a0 Evaluating and Solving a Tabular Function<\/h3>\r\n<p id=\"fs-id1165133356033\">Using <a class=\"autogenerated-content\" href=\"#Table_01_01_11\">Table 11<\/a>,<\/p>\r\n\r\n<ol id=\"fs-id1165137653327\" type=\"a\">\r\n \t<li>Evaluate [latex]g\\left(3\\right).[\/latex]<\/li>\r\n \t<li>Solve [latex]g\\left(n\\right)=6.[\/latex]<\/li>\r\n<\/ol>\r\n<table id=\"Table_01_01_11\" border=\"y\" summary=\"Two rows and six columns. The first row is labeled, \u201cn\u201d and the second row is labeled, \u201cg(n)\u201d. Reading the columns as ordered pairs, we have: (1, 8), (2, 6), (3, 7), (4, 6) , and (5, 8).\"><caption><strong>Table 11<\/strong><\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 134.775px;\"><strong>[latex]n[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1875px; text-align: center;\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 134.775px;\"><strong>[latex]g\\left(n\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">8<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 24.1875px; text-align: center;\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137748378\">[reveal-answer q=\"fs-id1165137748378\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137748378\"]\r\n<ol id=\"fs-id1165137725812\" type=\"a\">\r\n \t<li>Evaluating [latex]g\\left(3\\right)[\/latex] means determining the output value of the function [latex]g[\/latex] for the input value of [latex]n=3.[\/latex] The table output value corresponding to [latex]n=3[\/latex] is 7, so [latex]g\\left(3\\right)=7.[\/latex]<\/li>\r\n \t<li>Solving [latex]g\\left(n\\right)=6[\/latex] means identifying the input values, [latex]n,[\/latex] that produce an output value of 6. <a class=\"autogenerated-content\" href=\"#Table_01_01_12\">Table 11<\/a> shows two solutions: [latex]2[\/latex] and [latex]4.[\/latex]\u00a0 When we input 2 into the function [latex]g,[\/latex] our output is 6. When we input 4 into the function [latex]g,[\/latex] our output is also 6.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137584384\" class=\"precalculus tryit\">\r\n<h3>Try it #7<\/h3>\r\n<div id=\"ti_01_01_06\">\r\n<div id=\"fs-id1165137557816\">\r\n<p id=\"fs-id1165137557817\">Using <a class=\"autogenerated-content\" href=\"#Table_01_01_12\">Table 11<\/a>, evaluate [latex]g\\left(1\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137423936\">[reveal-answer q=\"fs-id1165137423936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137423936\"]\r\n<p id=\"fs-id1165137423937\">[latex]g\\left(1\\right)=8[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135696152\" class=\"bc-section section\">\r\n<h4>Finding Function Values from a Graph<\/h4>\r\n<p id=\"fs-id1165137779152\">Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\r\n\r\n<div id=\"Example_01_01_12\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134212105\">\r\n<div id=\"fs-id1165134212107\">\r\n<h3>Example 12:\u00a0 Reading Function Values from a Graph<\/h3>\r\n<p id=\"fs-id1165137469316\">Given the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_01_007\">Figure 7<\/a>,<\/p>\r\n\r\n<ol id=\"fs-id1165137604039\" type=\"a\">\r\n \t<li>Evaluate [latex]f\\left(2\\right).[\/latex]<\/li>\r\n \t<li>Solve [latex]f\\left(x\\right)=4.[\/latex]<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_3029\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3029\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142815\/985e7ba8c4bba18084e0450b429d6a0b53dfd707-1.jpeg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/> <strong>Figure 7<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_007\" class=\"small\">\r\n<div class=\"mceTemp\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137849160\">[reveal-answer q=\"fs-id1165137849160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137849160\"]\r\n<ol id=\"fs-id1165137871522\" type=\"a\">\r\n \t<li>To evaluate [latex]f\\left(2\\right),[\/latex] locate the point on the curve where [latex]x=2,[\/latex] then read the <em>y<\/em>-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right),[\/latex] so [latex]f\\left(2\\right)=1.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_01_008\">Figure 8<\/a>\r\n\r\n[caption id=\"attachment_3030\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142949\/2d4c4269758c5a422deed68865384d7b6774b924-1.jpeg\" alt=\"The same parabola as Figure 7, now with the point (2,1) labeled and the equation f(2)=1\" width=\"487\" height=\"445\" \/> <strong>Figure 8<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_008\" class=\"small\"><\/div><\/li>\r\n \t<li>To solve [latex]f\\left(x\\right)=4,[\/latex] we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4,[\/latex] we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right).[\/latex] These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]-1[\/latex] or [latex]3.[\/latex] This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4,[\/latex] or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_01_009\">Figure 9<\/a>.\r\n\r\n[caption id=\"attachment_3031\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3031\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143208\/4a94983cb7a344c2cba4c4a8bb3ed7e9f1f0b084-1.jpeg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/> <strong>Figure 9<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_009\" class=\"small\">\r\n<div class=\"mceTemp\"><\/div>\r\n<span id=\"fs-id1165137469773\">[\/hidden-answer]<\/span>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135149263\" class=\"precalculus tryit\">\r\n<h3>Try it #8<\/h3>\r\n<div id=\"ti_01_01_05\">\r\n<div id=\"fs-id1165137695207\">\r\n<p id=\"fs-id1165137695208\">Using <a class=\"autogenerated-content\" href=\"#Figure_01_01_007\">Figure 7<\/a>, solve [latex]f\\left(x\\right)=1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137598286\">[reveal-answer q=\"fs-id1165137598286\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137598286\"]\r\n<p id=\"fs-id1165137598287\">[latex]x=0[\/latex] or [latex]x=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135422920\" class=\"bc-section section\">\r\n<h3>Determining Whether a Function is One-to-One<\/h3>\r\n<p id=\"fs-id1165135678633\">Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in <a href=\"#figure10_1_add\">Figure 10<\/a>, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.<a id=\"figure10_1_add\"><\/a><\/p>\r\n\r\n\r\n[caption id=\"attachment_1600\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1600\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/18121430\/Bull-Market-Graph-Abramson-ch1-300x164.png\" alt=\"\" width=\"300\" height=\"164\" \/> Figure 10[\/caption]\r\n<p id=\"fs-id1165135245630\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in <a class=\"autogenerated-content\" href=\"#Table_01_01_13\">Table 12<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_01_13\" style=\"height: 132px;\" border=\"y\" summary=\"Two columns and five rows. The first column is labeled, \u201cLetter Grade\u201d, and the second column is labeled, \u201cGrade point average\u201d. Reading the rows as ordered pairs, we have: (A, 4.0), (B, 3.0), (C, 2.0), and (D, 1.0).\" width=\"200\"><caption><strong>Table 12<\/strong><\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<td class=\"border\">Letter grade<\/td>\r\n<td class=\"border\">Grade point average<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">A<\/td>\r\n<td class=\"border\">4.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">B<\/td>\r\n<td class=\"border\">3.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">C<\/td>\r\n<td class=\"border\">2.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">D<\/td>\r\n<td class=\"border\">1.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\r\n<p id=\"fs-id1165137628999\">To visualize this concept, let\u2019s look again at the two simple functions sketched in <a class=\"autogenerated-content\" href=\"#Figure_01_01_001\">Figure 1<\/a><strong>(a) <\/strong>and <a class=\"autogenerated-content\" href=\"#Figure_01_01_001\">Figure 1<\/a><strong>(b)<\/strong>. The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n.[\/latex] The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\r\n\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\nA <em><strong>one-to-one function<\/strong><\/em> is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n<div id=\"Example_01_01_13\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137755749\">\r\n<div id=\"fs-id1165137755752\">\r\n<h3>Example 13:\u00a0 Determining Whether a Relationship Is a One-to-One Function<\/h3>\r\n<p id=\"fs-id1165137892387\">Is the area of a circle a function of its radius? If yes, is the function one-to-one?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137892391\">[reveal-answer q=\"fs-id1165137892391\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137892391\"]\r\n<p id=\"fs-id1165134148470\">A circle of radius [latex]r[\/latex] has a unique area measure given by [latex]A=\\pi {r}^{2},[\/latex] so for any input, [latex]r,[\/latex] there is only one output, [latex]A.[\/latex] The area is a function of radius [latex]r.[\/latex]<\/p>\r\n<p id=\"fs-id1165137892326\">If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[\/latex] is given by the formula [latex]A=\\pi {r}^{2}.[\/latex] Because areas and radii are positive numbers, there is exactly one solution: [latex]\\sqrt{\\frac{A}{\\pi }}.[\/latex] So the area of a circle is a one-to-one function of the circle\u2019s radius.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137579363\" class=\"precalculus tryit\">\r\n<h3>Try it #9<\/h3>\r\n<div id=\"ti_01_01_10\">\r\n<div id=\"fs-id1165134079641\">\r\n<ol>\r\n \t<li>Is a balance a function of the bank account number?<\/li>\r\n \t<li>Is a bank account number a function of the balance?<\/li>\r\n \t<li>Is a balance a one-to-one function of the bank account number?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165137456018\">[reveal-answer q=\"fs-id1165137456018\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137456018\"]\r\n<ol>\r\n \t<li id=\"fs-id1165137592691\">Yes, because each bank account has a single balance at any given time;<\/li>\r\n \t<li>No, because several bank account numbers may have the same balance;<\/li>\r\n \t<li>No, because the same output may correspond to more than one input.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1327356\" class=\"precalculus tryit\">\r\n<h3>Try it #10<\/h3>\r\n<div id=\"ti_01_01_11\">\r\n<div id=\"fs-id1165137737576\">\r\n<p id=\"eip-id1166399990268\">Evaluate the following:<\/p>\r\n\r\n<ol id=\"fs-id1165135160248\" type=\"a\">\r\n \t<li>If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade? Explain.<\/li>\r\n \t<li>If so, is the function one-to-one? Explain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165137655289\">[reveal-answer q=\"fs-id1165137655289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137655289\"]\r\n<ol id=\"fs-id1165137655291\" type=\"a\">\r\n \t<li>Yes, the letter grade is a function of percent grade. Each input or percent grade is mapped to exactly one letter grade.<\/li>\r\n \t<li>No, it is not one-to-one. Each letter grade must be associated with more than one input.\u00a0 There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135435781\" class=\"bc-section section\">\r\n<h3>Using the Vertical Line Test<\/h3>\r\n<p id=\"fs-id1165135435786\">As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\r\n<p id=\"fs-id1165137637786\">Very often graphs name the input value [latex]x[\/latex] and the output value [latex]y,[\/latex] and we say [latex]y[\/latex] is a function of [latex]x,[\/latex] or [latex]y=f\\left(x\\right)[\/latex] when the function is named [latex]f.[\/latex] The graph of the function is the set of all points [latex]\\left(x,y\\right)[\/latex] in the plane that satisfies the equation [latex]y=f\\left(x\\right).[\/latex] If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em>x<\/em>-coordinate of each point is an input value and the <em>y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_01_011\">Figure 11<\/a> tell us that [latex]f\\left(0\\right)=2[\/latex] and [latex]f\\left(6\\right)=1.[\/latex] However, the set of all points [latex]\\left(x,y\\right)[\/latex] satisfying [latex]y=f\\left(x\\right)[\/latex] is a curve. The curve shown includes [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex] because the curve passes through those points.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3032\" align=\"aligncenter\" width=\"731\"]<img class=\"size-full wp-image-3032\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143447\/659b097af5e55cf05e8eba3f47d782bff5c352a7-2.jpeg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" \/> <strong>Figure 11<\/strong>[\/caption]\r\n<p id=\"fs-id1165137737620\">The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because a function can have only one output value for each input value. See <a class=\"autogenerated-content\" href=\"#Figure_01_01_012\">Figure 12<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3033\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143547\/73a7944f1a6253e4cc8726cc09e92801c34c9d76.jpeg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/> <strong>Figure 12<\/strong>[\/caption]\r\n\r\nThe second and third graphs in Figure 12 are not functions because the input value represented by the dotted line has two output values in each case.\u00a0 The two output values are the [latex]y[\/latex] values where each dotted line intersects the solid line.\u00a0 This contradicts the definition of a function. Remember, a function is a rule where each input is mapped to\u00a0<strong>exactly\u00a0<\/strong>one output.\r\n<div id=\"fs-id1165135460884\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137452182\"><strong>Given a graph, use the vertical line test to determine if the graph represents a function. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165133277614\" type=\"1\">\r\n \t<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If there is any such line, determine that the graph does not represent a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_01_14\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134541166\">\r\n<div id=\"fs-id1165137571591\">\r\n<h3>Example 14:\u00a0 Applying the Vertical Line Test<\/h3>\r\n<p id=\"fs-id1165137761111\">Which of the graphs in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a> represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_3034\" align=\"alignnone\" width=\"975\"]<img class=\"size-full wp-image-3034\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143659\/9102dcc3fb5ba81653f83c45e2a35606c7e74b44.jpeg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/> <strong>Figure 13<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135190052\">[reveal-answer q=\"fs-id1165135190052\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135190052\"]\r\n<p id=\"fs-id1165137629350\">If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a>. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at\u00a0<em>x<\/em>-values between -3 and 3, a vertical line would intersect the graph at more than one point, as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_016\">Figure 14<\/a>.\u00a0 This indicates that each of these inputs gets mapped to two different outputs.\u00a0 This contradicts the definition of a function, since we know a function maps each input to exactly one output.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3035\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3035\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143748\/4f55692043ccebf4409fbd35370e2d0aa41230cd.jpeg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/> <strong>Figure 14<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_016\" class=\"small\">\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134544969\" class=\"precalculus tryit\">\r\n<h3>Try it #11<\/h3>\r\n<div id=\"ti_01_01_04\">\r\n<div id=\"fs-id1165135600805\">\r\n<p id=\"fs-id1165135210137\">Does the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_01_017\">Figure 15<\/a> represent a function?<\/p>\r\n\r\n\r\n[caption id=\"attachment_3036\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3036\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143851\/6d3f6d402260a9c5f6e1312c594bd7b89db74e02.jpeg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/> <strong>Figure 15<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"small\"><\/div>\r\n<div id=\"ti_01_01_04\">\r\n<div id=\"fs-id1165134258608\">[reveal-answer q=\"fs-id1165134258608\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134258608\"]\r\n<p id=\"fs-id1165134258609\">Yes.\u00a0 Any vertical line will only pass through the graph once.\u00a0 This confirms the definition of one-to-one, since each output corresponds to only one input.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137610952\" class=\"bc-section section\">\r\n<h3>Using the Horizontal Line Test<\/h3>\r\n<p id=\"fs-id1165137871503\">Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.\u00a0 Each intersection along the horizontal line represents an x-value with the same output which contradicts the definition of one-to-one which states that each\u00a0 output value must be unique for the function to be one-to-one.<\/p>\r\n\r\n<div id=\"fs-id1165137736232\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165133437255\"><strong>Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137611853\" type=\"1\">\r\n \t<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If there is any such line, determine that the function is not one-to-one.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_01_15\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134389035\">\r\n<div id=\"fs-id1165134342668\">\r\n<h3>Example 15:\u00a0 Applying the Horizontal Line Test<\/h3>\r\n<p id=\"fs-id1165135434808\">Consider the functions shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13 <\/a><strong>(a)<\/strong> and <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a><strong>(b)<\/strong>. Are either of the functions one-to-one?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135521259\">[reveal-answer q=\"fs-id1165135521259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135521259\"]\r\n<p id=\"fs-id1165135185190\">The function in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a><strong>(a) <\/strong>is not one-to-one. The horizontal line shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_010\">Figure 16<\/a> intersects the graph of the function at two points.\u00a0 These two points have the same output value but different input values.\u00a0 Remember that the definition of a one-to-one function is that each output value corresponds to exactly one input value.\u00a0 Therefore, when a horizontal line intersects a graph at more than one point, we have a contradiction to this definition, and the function cannot be a one-to-one function.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3037\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3037 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14144007\/140fa74177d512d2a0de3741db0fb44d27334d60.jpeg\" alt=\"The horizontal line drawn intersects the polynomial two times.\" width=\"487\" height=\"445\" \/> <strong>Figure 16<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_01_010\" class=\"small\"><\/div>\r\n<p id=\"fs-id1165135151243\">The function in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a><strong>(b)<\/strong> is one-to-one. Any horizontal line will intersect a diagonal line at most once.\u00a0 This means that each output is associated with only one input value, satisfying the definition of a one-to-one function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135252051\" class=\"precalculus tryit\">\r\n<h3>Try it #12<\/h3>\r\n<div id=\"ti_01_01_12\">\r\n<div id=\"fs-id1165137749742\">\r\n<p id=\"fs-id1165137749744\">Is the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_016\">Figure 13<\/a> <strong>(c)<\/strong> one-to-one?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135255384\">[reveal-answer q=\"fs-id1165135255384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135255384\"]\r\n<p id=\"fs-id1165135255385\">No, because it does not pass the horizontal line test.\u00a0 We can see that there are horizontal lines that would intersect the graph in more than one point, indicating that an output value has more than one input value that corresponds to it.\u00a0 This contradicts the definition of one-to-one.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135545919\" class=\"bc-section section\">\r\n<h3>Identifying Basic Toolkit Functions<\/h3>\r\n<p id=\"fs-id1165137698132\">In this text, we will be exploring functions including the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our \u201ctoolkit functions,\u201d which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[\/latex] as the input variable and [latex]y=f\\left(x\\right)[\/latex] as the output variable.<\/p>\r\n<p id=\"fs-id1165135591070\">We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in <a class=\"autogenerated-content\" href=\"#Table_01_01_14\">Table 13<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_01_14\" style=\"height: 2048px;\" summary=\"The title is \u201cToolkit functions\u201d. There are three columns and ten rows. The first column is labeled, \u201cname\u201d, the second column is labeled, \u201cfunction\u201d, and the third column is labeled graph which contains pictures of the functions. The constant function is f(x) = c where c is the constant; the identity function is f(x) = x; the absolute function is f(x)=|x|; the quadratic function is f(x) = x^2; the cubic function is f(x)=x^3; the reciprocal function is f(x)=1\/x; the reciprocal squared function is f(x)=1\/x^2; the square root function is f(x)=sqrt(x); the cube root function is f(x) = x^(1\/3).\"><caption>Table 13<\/caption>\r\n<thead>\r\n<tr style=\"height: 74px;\">\r\n<td class=\"border\" style=\"text-align: center; width: 1006.5px; height: 74px;\" colspan=\"3\">\r\n<h1>Toolkit Functions<\/h1>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 110.5px;\">Name<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">Function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 517.5px;\">Graph<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 320px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Constant<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=c,[\/latex] where [latex]c[\/latex] is a constant<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137643159\"><img class=\"alignnone size-full wp-image-3067\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151148\/c166f7a1121575944ac02d2fc5c70770d0e963e6.jpeg\" alt=\"Graph of a constant function.\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 320px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Identity<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=x[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137811013\"><img class=\"alignnone wp-image-3068 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151218\/76a34ffd5fe5dc36af950679c61c54daeaf97c06.jpeg\" alt=\"Graph of a straight line, slope of 1.\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 320px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Absolute value<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165135195221\"><img class=\"alignnone size-full wp-image-3069\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151319\/ec22554e96a17ac92c66de3864e19c40780587dd.jpeg\" alt=\"Graph of an absolute value\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 320px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Quadratic<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137501903\"><img class=\"alignnone size-full wp-image-3070\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151357\/0f74efa823699938eb96f0ab6f767f6a545b525a.jpeg\" alt=\"Graph of a parabola\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 320px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Cubic<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137722123\"><img class=\"alignnone size-full wp-image-3071\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151429\/45dd3744b83abcb7676e82f77770ce461218c2e7.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 320px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Reciprocal<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165134544980\"><img class=\"alignnone size-full wp-image-3072\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151453\/0d2bcdc4558ed0b1afdbadc6e5b1c72b22985cdf.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 14px; width: 110.5px;\">Reciprocal squared<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 517.5px;\"><span id=\"fs-id1165137647610\"><img class=\"alignnone size-full wp-image-3073\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151527\/a33ee546ef2cbaa4a5ee7ee57c20463ec6801fff.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 14px; width: 110.5px;\">Square root<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 517.5px;\"><span id=\"fs-id1165137863670\"><img class=\"alignnone size-full wp-image-3074\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151553\/de470da28ca919df70b52178fd96577e7040eb51.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\" valign=\"top\">\r\n<td class=\"border\" style=\"height: 14px; width: 110.5px;\">Cube root<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 517.5px;\"><span id=\"fs-id1165137838612\"><img class=\"alignnone size-full wp-image-3075\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151627\/b7449e93ba1d9ebee82fbea8aea5676948ce9af9.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165134042311\" class=\"precalculus media\">\r\n<p id=\"fs-id1165135549046\">Access the following online resources for additional instruction and practice with functions.<\/p>\r\n\r\n<ul id=\"eip-id1165137846437\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/relationfunction\">Determine if a Relation is a Function<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=zT69oxcMhPw\r\n<ul id=\"eip-id1165137846437\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/vertlinetest\">Vertical Line Test<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=gO5WN9g1fJo&amp;feature=youtu.be%2F\r\n<ul id=\"eip-id1165137846437\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/introtofunction\">Introduction to Functions<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=sW9-zBeQpCU\r\n<ul id=\"eip-id1165137846437\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/vertlinegraph\">Vertical Line Test on Graph<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=5Z8DaZPJLKY\r\n<ul id=\"eip-id1165137846437\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/onetoone\">One-to-one Functions<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=QFOJmevha_Y&amp;feature=youtu.be%2F\r\n<ul id=\"eip-id1165137846437\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/graphonetoone\">Graphs as One-to-one Functions<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=tbSGdcSN8RE&amp;feature=youtu.be\/\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135203679\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165134393730\" style=\"height: 211px;\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Constant function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=c,[\/latex] where [latex]c[\/latex] is a constant<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Identity function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Absolute value function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Quadratic function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Cubic function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Reciprocal function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Reciprocal squared function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Square root function<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td class=\"border\" style=\"height: 44px; width: 159.5px;\">Cube root function<\/td>\r\n<td class=\"border\" style=\"height: 44px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 71px;\">\r\n<td class=\"border\" style=\"height: 71px; width: 159.5px;\">Difference Quotient<\/td>\r\n<td class=\"border\" style=\"height: 71px; width: 353.5px;\">[latex]\\frac{f(a+h)-f(a)}{h}\\\\[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137692068\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165137851183\">\r\n \t<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\r\n \t<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right).[\/latex]<\/li>\r\n \t<li>In tabular form, a function can be represented by rows or columns that relate to input and output values.<\/li>\r\n \t<li>To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.<\/li>\r\n \t<li>To solve for a specific function value, we determine the input values that yield the specific output value.<\/li>\r\n \t<li>An algebraic form of a function can be written from an equation.<\/li>\r\n \t<li>Input and output values of a function can be identified from a table.<\/li>\r\n \t<li>Relating input values to output values on a graph is another way to evaluate a function.<\/li>\r\n \t<li>A function is one-to-one if each output value corresponds to only one input value.<\/li>\r\n \t<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\r\n \t<li>The graph of a one-to-one function passes the horizontal line test.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137758543\">\r\n \t<dt>dependent variable<\/dt>\r\n \t<dd id=\"fs-id1165137758548\">an output variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137758552\">\r\n \t<dt>domain<\/dt>\r\n \t<dd id=\"fs-id1165137932576\">the set of all possible input values for a relation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137932580\">\r\n \t<dt>function<\/dt>\r\n \t<dd id=\"fs-id1165137932585\">a relation in which each input value yields a unique output value<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137932588\">\r\n \t<dt>horizontal line test<\/dt>\r\n \t<dd id=\"fs-id1165134149777\">a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134149782\">\r\n \t<dt>independent variable<\/dt>\r\n \t<dd id=\"fs-id1165134149787\">an input variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135511353\">\r\n \t<dt>input<\/dt>\r\n \t<dd id=\"fs-id1165135511359\">each object or value in a domain that relates to another object or value by a relationship known as a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135511364\">\r\n \t<dt>one-to-one function<\/dt>\r\n \t<dd id=\"fs-id1165135511369\">a function for which each value of the output is associated with a unique input value<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135508564\">\r\n \t<dt>output<\/dt>\r\n \t<dd id=\"fs-id1165135508569\">each object or value in the range that is produced when an input value is entered into a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135508573\">\r\n \t<dt>range<\/dt>\r\n \t<dd id=\"fs-id1165135315529\">the set of output values that result from the input values in a relation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135315533\">\r\n \t<dt>relation<\/dt>\r\n \t<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135315542\">\r\n \t<dt>vertical line test<\/dt>\r\n \t<dd id=\"fs-id1165134186374\">a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Determine and be able to explain whether a relation represents a function given a table or a graph.<\/li>\n<li>Evaluate functions and solve equations involving functions.<\/li>\n<li>Determine whether a function given numerically or graphically is one-to-one, and explain your rationale.<\/li>\n<li>Graph and name the functions listed in the library of functions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137431376\">A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.<\/p>\n<div id=\"fs-id1165133394710\" class=\"bc-section section\">\n<h3>Determining Whether a Relation Represents a Function<\/h3>\n<p id=\"fs-id1165137781542\">A relation is a set of ordered pairs. The set of the first components of each <span class=\"no-emphasis\">ordered pair<\/span> is called the <strong>domain <\/strong>and the set of the second components of each ordered pair is called the <strong>range<\/strong>. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.<\/p>\n<p id=\"fs-id1165137676332\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{\\left(1,\\text{ }2\\right),\\text{ }\\left(2,\\text{ }4\\right),\\text{ }\\left(3,\\text{ }6\\right),\\text{ }\\left(4,\\text{ }8\\right),\\text{ }\\left(5,\\text{ }10\\right)\\right\\}[\/latex]<\/p>\n<p class=\"unnumbered\" style=\"text-align: left;\">The domain is [latex]\\left\\{1,\\text{ }2,\\text{ }3,\\text{ }4,\\text{ }5\\right\\}.[\/latex] The range is [latex]\\left\\{2,\\text{ }4,\\text{ }6,\\text{ }8,\\text{ }10\\right\\}.[\/latex]<\/p>\n<p id=\"fs-id1165134234609\">Note that each value in the domain is also known as an <strong>input<\/strong> value, or independent variable, and is often labeled with the lowercase letter [latex]x.[\/latex] Each value in the range is also known as an <strong>output<\/strong> value, or dependent variable, and is often labeled lowercase letter [latex]y.[\/latex]<\/p>\n<p id=\"fs-id1165137748300\">A function [latex]f[\/latex] is a relation that assigns a single value in the range to each value in the domain<em>.<\/em> In other words, no [latex]x[\/latex]-values are repeated. For our example that relates the first five <span class=\"no-emphasis\">natural numbers<\/span> to numbers double their values, this relation is a function because each element in the domain, [latex]\\left\\{1,\\text{ }2,\\text{ }3,\\text{ }4,\\text{ }5\\right\\},[\/latex] is paired with exactly one element in the range, [latex]\\left\\{2,\\text{ }4,\\text{ }6,\\text{ }8,\\text{ }10\\right\\}.[\/latex]<\/p>\n<p id=\"fs-id1165135421564\">Now let\u2019s consider the set of ordered pairs that relates the terms \u201ceven\u201d and \u201codd\u201d to the first five natural numbers. It would appear as<\/p>\n<p id=\"fs-id1165133192963\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{\\left(\\text{odd},\\text{ }1\\right),\\text{ }\\left(\\text{even},\\text{ }2\\right),\\text{ }\\left(\\text{odd},\\text{ }3\\right),\\text{ }\\left(\\text{even},\\text{ }4\\right),\\text{ }\\left(\\text{odd},\\text{ }5\\right)\\right\\}[\/latex]<\/p>\n<p id=\"fs-id1165135419796\">Notice that each element in the domain, [latex]\\left\\{\\text{even,}\\text{ }\\text{odd}\\right\\}[\/latex] is <em>not<\/em> paired with exactly one element in the range, [latex]\\left\\{1,\\text{ }2,\\text{ }3,\\text{ }4,\\text{ }5\\right\\}.[\/latex] For example, the term \u201codd\u201d corresponds to three values from the range, [latex]\\left\\{1,\\text{ }3,\\text{ }5\\right\\}[\/latex] and the term \u201ceven\u201d corresponds to two values from the range, [latex]\\left\\{2,\\text{ }4\\right\\}.[\/latex] This violates the definition of a function, so this relation is not a function.<\/p>\n<p id=\"fs-id1165135176295\"><a class=\"autogenerated-content\" href=\"#Figure_01_01_001\">Figure 1<\/a> compares relations that are functions and not functions.<\/p>\n<div id=\"attachment_3023\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3023\" class=\"size-full wp-image-3023\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14135913\/d47bde493af93fa714bfc3dbe2a30008d2b7a5f01.jpeg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/><\/p>\n<p id=\"caption-attachment-3023\" class=\"wp-caption-text\"><strong>Figure 1:<\/strong> (a) This relationship is a function because each input is associated with a single output. Note that input [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n.[\/latex] (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input [latex]q[\/latex] is associated with two different outputs.<\/p>\n<\/div>\n<div id=\"fs-id1165137533627\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165135173375\">A <em><strong>function<\/strong><\/em> is a relation in which each possible input value leads to exactly one output value. We say \u201cthe output is a function of the input.\u201d<\/p>\n<p id=\"fs-id1165137661589\">The<strong> input values<\/strong> make up the <strong>domain<\/strong>, and the <strong>output values<\/strong> make up the <strong>range<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137445319\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137635406\"><strong>Given a relationship between two quantities, determine whether the relationship is a function.<\/strong><\/p>\n<ol id=\"fs-id1165134065124\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Identify the output values.<\/li>\n<li>If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_01_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137414052\">\n<div id=\"fs-id1165137559269\">\n<h3>Example 1:\u00a0 Determining If Menu Price Lists Are Functions<\/h3>\n<p id=\"fs-id1165137436464\">The coffee shop menu, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_004\">Figure 2<\/a> consists of items and their prices.<\/p>\n<ol id=\"fs-id1165137646341\" type=\"a\">\n<li>Is price a function of the item?<\/li>\n<li>Is the item a function of the price?<\/li>\n<\/ol>\n<div id=\"attachment_3024\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3024\" class=\"size-full wp-image-3024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142007\/628598f20aa79562d901ad8842b1a402adfeb5c2.jpeg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is 1.49 and a jelly donut and chocolate donut are 1.99.\" width=\"487\" height=\"233\" \/><\/p>\n<p id=\"caption-attachment-3024\" class=\"wp-caption-text\"><strong>Figure 2<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_004\" class=\"small\">\n<div class=\"mceTemp\"><\/div>\n<\/div>\n<div><\/div>\n<\/div>\n<div id=\"fs-id1165135419802\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135419802\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135419802\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137643241\" type=\"a\">\n<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices. See <a class=\"autogenerated-content\" href=\"#Figure_01_01_027\">Figure 3<\/a>.\n<div id=\"attachment_3025\" style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3025\" class=\"size-full wp-image-3025\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142202\/472edeec5bc2bfb61746e8c5b3eedf15d73f1c02.jpeg\" alt=\"A menu of donut prices from a coffee shop where a plain donut is 1.49 and a jelly donut and chocolate donut are 1.99.\" width=\"731\" height=\"241\" \/><\/p>\n<p id=\"caption-attachment-3025\" class=\"wp-caption-text\"><strong>Figure 3<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_027\" class=\"medium\">\n<div class=\"mceTemp\"><\/div>\n<\/div>\n<p id=\"fs-id1165135532324\">Each item on the menu has only one price, so the price is a function of the item.<\/p>\n<\/li>\n<li>Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See <a class=\"autogenerated-content\" href=\"#Figure_01_01_028\">Figure 4<\/a>.\n<p id=\"fs-id1165137754835\">Therefore, the item is a not a function of price.<\/p>\n<\/li>\n<\/ol>\n<div id=\"attachment_3026\" style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3026\" class=\"wp-image-3026 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142259\/3adba7b02c540ad82f7d8fa53691f3d781923d0f.jpeg\" alt=\"Association of the prices to the donuts.\" width=\"731\" height=\"241\" \/><\/p>\n<p id=\"caption-attachment-3026\" class=\"wp-caption-text\"><strong>Figure 4<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_01_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137437773\">\n<div id=\"fs-id1165135620873\">\n<h3>Example 2:\u00a0 Determining If Class Grade Rules Are Functions<\/h3>\n<p id=\"fs-id1165137442099\">In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? <a class=\"autogenerated-content\" href=\"#Table_01_01_01\">Table 1<\/a> shows a possible rule for assigning grade points.<\/p>\n<table id=\"Table_01_01_01\" class=\"lines\" style=\"height: 24px;\" summary=\"Title of the table is \u201cClass Grades\u201d. It contains two columns and ten rows. The first column is labeled, \u201cPercent Grade\u201d, and the second column is labeled, \u201cGrade point average\u201d. Reading the rows as ordered pairs, we have: (92-100, 4.0), (87-91, 3.5), (78-86, 3.0), (72-77, 2.5), (67-71, 2.0), (62-66, 1.5), (57-61, 1.0), and (0-56, 0.0).\">\n<caption><strong>Table 1<\/strong><\/caption>\n<tbody>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 114px; height: 12px;\"><strong>Percent grade<\/strong><\/td>\n<td class=\"border\" style=\"width: 28px; height: 12px; text-align: center;\">0\u201356<\/td>\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">57\u201361<\/td>\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">62\u201366<\/td>\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">67\u201371<\/td>\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">72\u201377<\/td>\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">78\u201386<\/td>\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">87\u201391<\/td>\n<td class=\"border\" style=\"width: 42px; text-align: center; height: 12px;\">92\u2013100<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 114px; height: 12px;\"><strong>Grade point average<\/strong><\/td>\n<td class=\"border\" style=\"width: 28px; text-align: center; height: 12px;\">0.0<\/td>\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">1.0<\/td>\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">1.5<\/td>\n<td class=\"border\" style=\"width: 35px; height: 12px; text-align: center;\">2.0<\/td>\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">2.5<\/td>\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">3.0<\/td>\n<td class=\"border\" style=\"width: 35px; text-align: center; height: 12px;\">3.5<\/td>\n<td class=\"border\" style=\"width: 42px; text-align: center; height: 12px;\">4.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div><\/div>\n<div id=\"fs-id1165135424616\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135424616\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135424616\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135260743\">For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.<\/p>\n<p id=\"fs-id1165137807321\">In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137588587\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_01_01_01\">\n<div id=\"fs-id1165135667843\">\n<p id=\"fs-id1165137627634\"><a class=\"autogenerated-content\" href=\"#Table_01_01_02\">Table 2<\/a><a class=\"footnote\" title=\"http:\/\/www.baseball-almanac.com\/legendary\/lisn100.shtml. Accessed 3\/24\/2014\" id=\"return-footnote-26-1\" href=\"#footnote-26-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0lists the five greatest baseball players of all time in order of rank.<\/p>\n<table id=\"Table_01_01_02\" class=\"lines\" style=\"width: 200px; height: 84px;\" summary=\"Six rows and two columns. The first column is labeled, \u201cplayer name\u201d, and the second column is labeled, \u201crank\u201d. Reading the rows as ordered pairs, we have: (Babe Ruth, 1), (Willie Mays, 2), (Ty Cobb, 3), (Walter Johnson, 4), and (Hank Aaron, 5).\" cellpadding=\"0\">\n<caption><strong>Table 2<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr style=\"height: 14px;\">\n<td class=\"border\">Player<\/td>\n<td class=\"border\">Rank<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td class=\"border\">Babe Ruth<\/td>\n<td class=\"border\">1<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\">Willie Mays<\/td>\n<td class=\"border\">2<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\">Ty Cobb<\/td>\n<td class=\"border\">3<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\">Walter Johnson<\/td>\n<td class=\"border\">4<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\">Hank Aaron<\/td>\n<td class=\"border\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1165137501241\" type=\"a\">\n<li>Is the rank a function of the player name?<\/li>\n<li>Is the player name a function of the rank?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137724415\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137724415\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137724415\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137682011\">a. Yes.\u00a0 The input would be the player name, and the output would be the rank.\u00a0 Each player is mapped to exactly one rank.\u00a0 This meets the definition of function.<\/p>\n<p>b. Yes.\u00a0 The input would be the rank, and the output would be the player name.\u00a0 Each rank is mapped to exactly one name, so this meets the definition of function.\u00a0 However, if two players had been tied for, say, 4th place, then the name would not have been a function of rank.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134474160\" class=\"bc-section section\">\n<h4>Using Function Notation<\/h4>\n<p id=\"fs-id1165133359348\">Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <span class=\"no-emphasis\">function notation<\/span> is one representation that facilitates working with functions.<\/p>\n<p id=\"fs-id1165137453971\">To represent \u201cheight is a function of age,\u201d we start by identifying the descriptive variables [latex]h[\/latex] for height and [latex]a[\/latex] for age. The letters [latex]f,\\text{ }g,[\/latex] and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,\\text{ }y,[\/latex] and [latex]z[\/latex] to represent numbers and [latex]A,\\text{ }B,[\/latex] and [latex]C[\/latex] to represent sets.<\/p>\n<p id=\"fs-id1165135332760\" class=\"unnumbered\" style=\"padding-left: 30px;\">[latex]\\begin{array}{lllll}h\\text{ is }f\\text{ of }a\\hfill & \\hfill & \\hfill & \\hfill & \\text{We name the function }f;\\text{ height is a function of age}.\\hfill \\\\ h=f\\left(a\\right)\\hfill & \\hfill & \\hfill & \\hfill & \\text{We use parentheses to indicate the function input}\\text{. }\\hfill \\\\ f\\left(a\\right)\\hfill & \\hfill & \\hfill & \\hfill & \\text{We name the function }f;\\text{ the expression is read as \u201c}f\\text{ of }a\\text{.\u201d}\\hfill \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137766965\">Remember, we can use any letter to name the function; the notation [latex]f\\left(a\\right)[\/latex] shows us that height, [latex]h,[\/latex] depends on age, [latex]a.[\/latex] The value [latex]a[\/latex] must be put into the function [latex]f[\/latex] to get the height. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\n<p id=\"fs-id1165135436660\">We can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means \u201cfirst add <em>a<\/em> and <em>b<\/em>, and the result is the input for the function <em>f<\/em>.\u201d The operations must be performed in this order to obtain the correct result.<\/p>\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p><em><strong>function notation<\/strong><\/em>:\u00a0\u00a0 The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f.[\/latex] This is read as [latex]\u201cy[\/latex] is a function of [latex]x.\u201d[\/latex] The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y\\text{, }[\/latex] or [latex]f\\left(x\\right),[\/latex] represents the output value, or dependent variable.<\/p>\n<\/div>\n<div id=\"Example_01_01_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135612059\">\n<div id=\"fs-id1165135705803\">\n<h3>Example 3:\u00a0 Using Function Notation for Days in a Month<\/h3>\n<p id=\"fs-id1165137757351\">Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.<\/p>\n<\/div>\n<div id=\"fs-id1165137405547\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137405547\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137405547\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137657617\">The number of days in a month is a function of the name of the month, so if we name the function [latex]f,[\/latex] we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex] or [latex]d=f\\left(m\\right).[\/latex] The name of the month is the input to a \u201crule\u201d that associates a specific number (the output) with each input.<\/p>\n<div id=\"attachment_3027\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3027\" class=\"size-full wp-image-3027\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142437\/3934b41db8099eaa597af21c2abc98e820d55d15.jpeg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" \/><\/p>\n<p id=\"caption-attachment-3027\" class=\"wp-caption-text\"><strong>Figure 5<\/strong><\/p>\n<\/div>\n<div id=\"Image_01_01_005\" class=\"unnumbered\"><\/div>\n<p id=\"fs-id1165135417826\">For example, [latex]f\\left(\\text{March}\\right)=31,[\/latex] because March has 31 days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).<\/p>\n<h3>Analysis<\/h3>\n<p>Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers or algebraic expressions as inputs and outputs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_01_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137441910\">\n<div id=\"fs-id1165137527239\">\n<h3>Example 4:\u00a0 Interpreting Function Notation<\/h3>\n<p id=\"fs-id1165137526811\">A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N,[\/latex] in a town in year [latex]y.[\/latex] What does [latex]f\\left(2005\\right)=300[\/latex] represent?<\/p>\n<\/div>\n<div id=\"fs-id1165137834021\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137834021\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137834021\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137424675\">When we read [latex]f\\left(2005\\right)=300,[\/latex] we see that the input year is 2005. The value for the output, the number of police officers, [latex]N,[\/latex] is 300. Remember, [latex]N=f\\left(y\\right).[\/latex] The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were 300 police officers in the town.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134257606\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"fs-id1165137564344\">\n<div id=\"fs-id1165137564345\">\n<p id=\"fs-id1165137596424\">Use function notation to express the weight of a pig in pounds as a function of its age in days [latex]d\\text{.}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137871618\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137871618\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137871618\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137619935\">[latex]w=f\\left(d\\right)[\/latex] since\u00a0 the input would be days and the output would be weight.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137740780\" class=\"precalculus qa key-takeaways\">\n<h3>Q&amp;A<\/h3>\n<p id=\"eip-id1165132005171\"><strong>Instead of a notation such as [latex]y=f\\left(x\\right),[\/latex] could we use the same symbol for the output as for the function, such as [latex]y=y\\left(x\\right),[\/latex] meaning \u201c<em>y<\/em> is a function of <em>x<\/em>?\u201d<\/strong><\/p>\n<p id=\"fs-id1165137605080\"><em>Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as [latex]f,[\/latex] which is a rule or procedure, and the output [latex]y[\/latex] we get by applying [latex]f[\/latex] to a particular input [latex]x.[\/latex] This is why we usually use notation such as [latex]y=f\\left(x\\right),P=W\\left(d\\right),[\/latex] and so on.<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137804204\" class=\"bc-section section\">\n<h4>Representing Functions Using Tables<\/h4>\n<p id=\"fs-id1165137648317\">A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.<\/p>\n<p id=\"fs-id1165137761188\"><a class=\"autogenerated-content\" href=\"#Table_01_01_03\">Table 3<\/a> lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]f[\/latex] where [latex]D=f\\left(m\\right)[\/latex] identifies months by an integer rather than by name.<\/p>\n<table id=\"Table_01_01_03\" class=\"lines\" summary=\"Two rows and thirteen columns. The first row is labeled, \u201c(input) Month number, m\u201d and the second row is labeled, \u201c(output) Days in months, D\u201d. Reading the columns as ordered pairs, we have: (1, 31), (2, 28), (3, 31), (4, 30), (5, 31), (6, 30), (7, 31), (8, 31), (9, 30) , (10, 31), (11, 30), and (12, 31).\">\n<caption><strong>Table 3<\/strong><\/caption>\n<tbody>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"height: 14px; width: 129.017px;\"><strong>Month number, [latex]m[\/latex](input)<\/strong><\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">4<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">5<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">8<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">9<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">10<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">11<\/td>\n<td class=\"border\" style=\"height: 14px; width: 42.45px; text-align: center;\">12<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"height: 14px; width: 129.017px;\"><strong>Days in month, [latex]D[\/latex](output)<\/strong><\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">31<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.05px; text-align: center;\">28<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">31<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">31<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0667px; text-align: center;\">31<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">31<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0833px; text-align: center;\">31<\/td>\n<td class=\"border\" style=\"height: 14px; width: 31.0333px; text-align: center;\">30<\/td>\n<td class=\"border\" style=\"height: 14px; width: 42.45px; text-align: center;\">31<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135191568\"><a class=\"autogenerated-content\" href=\"#Table_01_01_04\">Table 4<\/a> defines a function [latex]Q=g\\left(n\\right).[\/latex] Remember, this notation tells us that [latex]g[\/latex] is the name of the function that takes the input [latex]n[\/latex] and gives the output [latex]Q\\text{ .}[\/latex]<\/p>\n<table id=\"Table_01_01_04\" style=\"height: 51px; width: 335px;\" summary=\"Two rows and six columns. The first row is labeled, \u201cn\u201d and the second row is labeled, \u201cQ\u201d. Reading the columns as ordered pairs, we have: (1, 8), (2, 6), (3, 7), (4, 6) , and (5, 8).\">\n<caption><strong>Table 4<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"height: 14px; width: 28.183px;\">[latex]n[\/latex]<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2167px; text-align: center;\">4<\/td>\n<td class=\"border\" style=\"height: 14px; width: 23.2px; text-align: center;\">5<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"height: 14px; width: 28.183px;\">[latex]Q[\/latex]<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">8<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2333px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"height: 14px; width: 28.2167px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"height: 14px; width: 23.2px; text-align: center;\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137561574\"><a class=\"autogenerated-content\" href=\"#Table_01_01_05\">Table 5<\/a> displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.<\/p>\n<table id=\"Table_01_01_05\" style=\"width: 523px;\" summary=\"Two rows and eight columns. The first row is labeled, \u201c(input) a, age in years\u201d and the second row is labeled, \u201c(output) h, height in inches\u201d. Reading the columns as ordered pairs, we have: (5, 40), (5, 42) , (6, 44), (7, 47), (8, 50), (9, 52), and (10, 54).\">\n<caption><strong>Table 5<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>Age in years, [latex]a[\/latex] (input)<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<td class=\"border\" style=\"text-align: center;\">9<\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>Height in inches,[latex]h[\/latex] (output)<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">40<\/td>\n<td class=\"border\" style=\"text-align: center;\">42<\/td>\n<td class=\"border\" style=\"text-align: center;\">44<\/td>\n<td class=\"border\" style=\"text-align: center;\">47<\/td>\n<td class=\"border\" style=\"text-align: center;\">50<\/td>\n<td class=\"border\" style=\"text-align: center;\">52<\/td>\n<td class=\"border\" style=\"text-align: center;\">54<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1165137804163\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134200185\"><strong>Given a table of input and output values, determine whether the table represents a function. <\/strong><\/p>\n<ol id=\"fs-id1165137461155\" type=\"1\">\n<li>Identify the input and output values.<\/li>\n<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_01_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137416794\">\n<div id=\"fs-id1165135591087\">\n<h3>Example 5:\u00a0 Identifying Tables that Represent Functions<\/h3>\n<p id=\"fs-id1165135503697\">Which table, <a class=\"autogenerated-content\" href=\"#Table_01_01_06\">Table 6<\/a>, <a class=\"autogenerated-content\" href=\"#Table_01_01_07\">Table 7<\/a>, or <a class=\"autogenerated-content\" href=\"#Table_01_01_08\">Table 8<\/a>, represents a function (if any)?<\/p>\n<table id=\"Table_01_01_06\" style=\"width: 150px; height: 96px; width: 150px;\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (2, 1), (5, 3), and (8, 6).\">\n<caption><strong>Table 6<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">Input<\/td>\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">Output<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">2<\/td>\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">5<\/td>\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.9px;\">8<\/td>\n<td class=\"border\" style=\"height: 12px; width: 203.117px;\">6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"Table_01_01_07\" style=\"width: 150px; height: 96px;\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (-3, 5), (0, 1), and (4, 5).\">\n<caption><strong>Table 7<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<td class=\"border\">Input<\/td>\n<td class=\"border\">Output<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\">\u20133<\/td>\n<td class=\"border\">5<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">0<\/td>\n<td class=\"border\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">4<\/td>\n<td class=\"border\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"Table_01_01_08\" style=\"width: 150px; height: 96px;\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (1, 0), (5, 2), and (5, 4).\">\n<caption><strong>Table 8<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th class=\"border\">Input<\/th>\n<th class=\"border\">Output<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\">1<\/td>\n<td class=\"border\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">5<\/td>\n<td class=\"border\">2<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">5<\/td>\n<td class=\"border\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137665675\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137665675\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137665675\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137401396\"><a class=\"autogenerated-content\" href=\"#Table_01_01_06\">Table 6<\/a> and <a class=\"autogenerated-content\" href=\"#Table_01_01_07\">Table 7<\/a> define functions. In both, each input value corresponds to exactly one output value. <a class=\"autogenerated-content\" href=\"#Table_01_01_08\">Table 8<\/a> does not define a function because the input value of 5 corresponds to two different output values.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>When a table represents a function, corresponding input and output values can also be specified using function notation.<\/p>\n<p>The function represented by <a class=\"autogenerated-content\" href=\"#Table_01_01_06\">Table 6<\/a> can be represented by writing<\/p>\n<p id=\"fs-id1165137404863\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(2\\right)=1,\\text{ }f\\left(5\\right)=3,[\/latex] and [latex]f\\left(8\\right)=6.[\/latex]<\/p>\n<p id=\"fs-id1165137619677\">Similarly, the statements<\/p>\n<p id=\"fs-id1165137589116\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(-3\\right)=5,\\text{ }g\\left(0\\right)=1,[\/latex] and [latex]g\\left(4\\right)=5[\/latex]<\/p>\n<p id=\"fs-id1165137715365\">represent the function in <a class=\"autogenerated-content\" href=\"#Table_01_01_07\">Table 7<\/a>.<\/p>\n<p id=\"fs-id1165137656795\"><a class=\"autogenerated-content\" href=\"#Table_01_01_08\">Table 8<\/a> cannot be expressed in a similar way because it does not represent a function.<\/p>\n<div id=\"fs-id1165137749258\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"ti_01_01_03\">\n<div id=\"fs-id1165137698328\">\n<p id=\"fs-id1165137698329\">Does <a class=\"autogenerated-content\" href=\"#Table_01_01_09\">Table 9<\/a> represent a function?<\/p>\n<table id=\"Table_01_01_09\" style=\"height: 96px; width: 276px; width: 150px;\" summary=\"Four rows and two columns. The first column is labeled, \u201cinput\u201d, and the second column is labeled, \u201coutput\u201d. Reading the rows as ordered pairs, we have: (1, 10), (2, 100), and (3, 1000).\">\n<caption><strong>Table 9<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"text-align: center;\">Input<\/td>\n<td class=\"border\" style=\"text-align: center;\">Output<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">100<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\n<td class=\"border\" style=\"text-align: center;\">1000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1165135322022\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135322022\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135322022\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137844279\">Yes.\u00a0 Each input corresponds to exactly one output.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137503241\" class=\"bc-section section\">\n<h3>Finding Input and Output Values of a Function<\/h3>\n<p id=\"fs-id1165137470651\">When we know an input value and want to determine the corresponding output value for a function, we <strong><em>evaluate<\/em><\/strong> the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.<\/p>\n<p id=\"fs-id1165137735634\">When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function\u2019s formula and <strong><em>solve<\/em><\/strong> for the input. Solving can produce more than one solution because different input values can produce the same output value.<\/p>\n<div id=\"fs-id1165137425943\" class=\"bc-section section\">\n<h4>Evaluation of Functions in Algebraic Forms<\/h4>\n<p id=\"fs-id1165137655584\">When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\\left(x\\right)=5-3{x}^{2}[\/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.<\/p>\n<div id=\"fs-id1165135613610\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137767182\"><strong>Given the formula for a function, evaluate. <\/strong><\/p>\n<ol id=\"fs-id1165137629040\" type=\"1\">\n<li>Replace the input variable in the formula with the value provided.<\/li>\n<li>Calculate the result.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_01_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137742220\">\n<div id=\"fs-id1165137455592\">\n<h3>Example 6:\u00a0 Evaluating Functions at Specific Values<\/h3>\n<p id=\"fs-id1165134193005\">Evaluate [latex]f\\left(x\\right)={x}^{2}+3x-4[\/latex] at<\/p>\n<ol id=\"fs-id1165137648008\" type=\"a\">\n<li>[latex]2[\/latex]<\/li>\n<li>[latex]a[\/latex]<\/li>\n<li>[latex]a+h[\/latex]<\/li>\n<li>[latex]\\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135397244\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135397244\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135397244\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137936905\">Replace the [latex]x[\/latex] in the function with each specified value.<\/p>\n<ol id=\"fs-id1165137778273\" type=\"a\">\n<li>Because the input value is a number, 2, we can use simple algebra to simplify.\n<p id=\"fs-id1165135160774\" class=\"unnumbered hanging-indent\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(2\\right)&={2}^{2}+3\\left(2\\right)-4\\\\&=4+6-4\\\\&=6\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\n<\/li>\n<li>In this case, the input value is a letter so we cannot simplify the answer any further.\n<p id=\"fs-id1165137638318\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a-4[\/latex][latex]\\\\[\/latex]<\/p>\n<\/li>\n<li>With an input value of[latex]\\text{ }a+h,[\/latex] we must use the distributive property.\n<p id=\"fs-id1165137911654\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(a+h\\right)&={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4\\\\&={a}^{2}+2ah+{h}^{2}+3a+3h-4\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: center;\">In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\n<p id=\"fs-id1165135154122\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h-4[\/latex]<\/p>\n<p id=\"fs-id1165135632109\">and we know that<\/p>\n<p id=\"fs-id1165137471110\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a-4.[\/latex]<\/p>\n<p id=\"fs-id1165137767461\">Now we combine the results and simplify.<\/p>\n<p id=\"fs-id1165137573884\" class=\"unnumbered\" style=\"text-align: left;\">[latex]\\frac{f(a+h)-f(a)}{h}\\\\[\/latex]<\/p>\n<p>[latex]\\begin{align*}\\text{ }\\text{ }\\text{ }\\text{ }&=\\frac{({{a}^{2}}+2ah+{{h}^{2}}+3a+3h-4)-({{a}^{2}}+3a-4)}{h}&&\\text{ }\\\\&=\\frac{2ah+{{h}^{2}}+3h}{h}\\\\&=\\frac{h(2a+h+3)}{h}&& \\text{Factor out h.} \\\\&=2a+h+3&&\\text{Simplify.} \\\\ \\end{align*}[\/latex]\n<\/p><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In Example 6, you worked with the expression below:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align*}\\frac{f(a+h)-f(a)}{h}\\\\\\end{align*}[\/latex]<\/p>\n<p style=\"text-align: left;\">This is called the difference quotient. In Calculus, we use the difference quotient to develop an important concept called the derivative. You should work to become very comfortable with the difference quotient. We will work with it more in later sections of this chapter.<\/p>\n<div id=\"Example_01_01_07\" class=\"textbox examples\">\n<div id=\"fs-id1165134043756\">\n<div id=\"fs-id1165137705537\">\n<h3>Example 7:\u00a0 Evaluating Functions<\/h3>\n<p id=\"fs-id1165137731385\">Given the function [latex]h\\left(p\\right)={p}^{2}+2p,[\/latex] evaluate [latex]h\\left(4\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137433651\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137433651\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137433651\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137433653\">To evaluate [latex]h\\left(4\\right),[\/latex] we substitute the value 4 for the input variable [latex]p[\/latex] in the given function.<\/p>\n<p id=\"fs-id1165137444745\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}h\\left(p\\right)&={p}^{2}+2p\\\\h\\left(4\\right)&={\\left(4\\right)}^{2}+2\\left(4\\right)\\\\ &=16+8\\\\&=24\\end{align*}[\/latex]<\/p>\n<p id=\"fs-id1165137785006\">Therefore, for an input of 4, we have an output of 24.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137704746\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_01_01_08\">\n<div id=\"fs-id1165134039322\">\n<p id=\"fs-id1165134039323\">Given the function [latex]g\\left(m\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{m-4},[\/latex] evaluate [latex]g\\left(5\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137441862\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137441862\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137441862\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134037488\">[latex]g\\left(5\\right)=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_01_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137459747\">\n<div id=\"fs-id1165137459749\">\n<h3>Example 8:\u00a0 Solving Functions<\/h3>\n<p id=\"fs-id1165137460826\">Given the function [latex]h\\left(p\\right)={p}^{2}+2p,[\/latex] solve for [latex]h\\left(p\\right)=3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132971707\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132971707\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132971707\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165135195145\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}h\\left(p\\right)&=3&&\\text{ }\\\\{p}^{2}+2p&=3&&\\text{Substitute the original function.}\\\\{p}^{2}+2p-3&=0&&\\text{Subtract 3 from each side}.\\\\\\left(p+3\\text{)(}p-1\\right)&=0&&\\text{Factor}.\\end{align*}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137770370\">If [latex]\\left(p+3\\right)\\left(p-1\\right)=0,[\/latex] either [latex]\\left(p+3\\right)=0[\/latex] or [latex]\\left(p-1\\right)=0[\/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[\/latex] in each case.<\/p>\n<div id=\"fs-id1165134114001\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}\\left(p+3\\right)=0,\\hfill & p=-3\\hfill \\\\ \\left(p-1\\right)=0,\\hfill & p=1\\hfill \\end{array}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165134468906\">This gives us two solutions. The output [latex]h\\left(p\\right)=3[\/latex] when the input is either [latex]p=1[\/latex] or [latex]p=-3.[\/latex] We can also verify by graphing as in <a class=\"autogenerated-content\" href=\"#Figure_01_01_006\">Figure 6<\/a>. The graph verifies that [latex]h\\left(1\\right)=h\\left(-3\\right)=3[\/latex] and [latex]h\\left(4\\right)=24.[\/latex]<\/p>\n<div id=\"attachment_3028\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3028\" class=\"size-full wp-image-3028\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142655\/5f6fe9b75ac7f991d78dac5f26509ecb4d659b86.jpeg\" alt=\"Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).\" width=\"487\" height=\"459\" \/><\/p>\n<p id=\"caption-attachment-3028\" class=\"wp-caption-text\"><strong>Figure 6<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_006\" class=\"small\">\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133056923\" class=\"precalculus tryit\">\n<h3>Try it #5<\/h3>\n<div id=\"ti_01_01_09\">\n<div id=\"fs-id1165134170173\">\n<p id=\"fs-id1165134170174\">Given the function [latex]g\\left(m\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{m-4},[\/latex] solve [latex]g\\left(m\\right)=2.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135664055\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135664055\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135664055\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135664056\">[latex]m=8[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137591827\" class=\"bc-section section\">\n<h4>Evaluating Functions Expressed in Formulas<\/h4>\n<p id=\"fs-id1165137598337\">Some functions are defined by mathematical rules or procedures expressed in <span class=\"no-emphasis\">equation<\/span> form. If it is possible to express the function output with a <span class=\"no-emphasis\">formula<\/span> involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[\/latex] expresses a functional relationship between [latex]n[\/latex] and [latex]p.[\/latex] We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n.[\/latex]<\/p>\n<div id=\"fs-id1165137827882\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165132034236\"><strong>Given a function in equation form, write its algebraic formula.<\/strong><\/p>\n<ol id=\"fs-id1165134544989\" type=\"1\">\n<li>Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves <em>only<\/em> the input variable.<\/li>\n<li>Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_01_09\" class=\"textbox examples\">\n<div id=\"fs-id1165137634899\">\n<div id=\"fs-id1165137560474\">\n<h3>Example 9:\u00a0 Finding an Equation of a Function<\/h3>\n<p id=\"fs-id1165137452465\">Express the relationship [latex]2n+6p=12[\/latex] as a function [latex]p=f\\left(n\\right),[\/latex] if possible.<\/p>\n<\/div>\n<div id=\"fs-id1165137832865\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137832865\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137832865\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137832868\">To express the relationship in this form, we need to be able to write the relationship where [latex]p[\/latex] is a function of [latex]n,[\/latex] which means writing it as [latex]p=\\left[\\text{expression}\\text{ }\\text{involving}\\text{ }n\\right].[\/latex]<\/p>\n<div id=\"fs-id1165131920640\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}2n+6p&=12&&\\text{ }\\\\6p&=12-2n&&\\text{Subtract }2n\\text{ from both sides}.\\\\p&=\\frac{12-2n}{6}&&\\text{Divide both sides by 6 and simplify}.\\\\p&=\\frac{12}{6}-\\frac{2n}{6}&&\\text{ } \\\\p&=2-\\frac{1}{3}n&&\\text{ }\\end{align*}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135513733\">Therefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as<\/p>\n<div id=\"fs-id1165135187787\" class=\"unnumbered\" style=\"text-align: center;\">[latex]p=f\\left(n\\right)=2-\\frac{1}{3}n.[\/latex]<\/div>\n<h3>Analysis<\/h3>\n<div>It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_01_10\" class=\"textbox examples\">\n<div id=\"fs-id1165135378843\">\n<div id=\"fs-id1165135378845\">\n<h3>Example 10:\u00a0 Expressing the Equation of a Circle as a Function<\/h3>\n<p id=\"fs-id1165137758151\">Does the equation [latex]{x}^{2}+{y}^{2}=1[\/latex] represent a function with [latex]x[\/latex] as input and [latex]y[\/latex] as output? If so, express the relationship as a function [latex]y=f\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135424684\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135424684\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135424684\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137937536\">First we subtract [latex]{x}^{2}[\/latex] from both sides.<\/p>\n<div id=\"fs-id1165134054911\" class=\"unnumbered\" style=\"text-align: center;\">[latex]{y}^{2}=1-{x}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165133437258\">We now try to solve for [latex]y[\/latex] in this equation.<\/p>\n<div id=\"fs-id1165137416396\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}y=\u00b1\\sqrt{1-{x}^{2}}\\hfill \\\\ \\text{ }=+\\sqrt{1-{x}^{2}}\\text{ and }-\\sqrt{1-{x}^{2}}\\hfill \\end{array}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135369156\">We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137647659\" class=\"precalculus tryit\">\n<h3>Try it #6<\/h3>\n<div id=\"fs-id1165137698725\">\n<div id=\"fs-id1165137698726\">\n<p id=\"fs-id1165137698727\">If [latex]x-8{y}^{3}=0,[\/latex] express [latex]y[\/latex] as a function of [latex]x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137677194\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137677194\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137677194\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135344102\">[latex]y=f\\left(x\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2}3]{x}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135581166\" class=\"precalculus qa key-takeaways\">\n<h3>Q&amp;A<\/h3>\n<p id=\"eip-id1165135547539\"><strong>Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?<\/strong><\/p>\n<p><em>Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y},[\/latex] if we want to express [latex]y[\/latex] as a function of [latex]x,[\/latex] there is no simple algebraic formula involving only [latex]x[\/latex] that equals [latex]y.[\/latex] However, each [latex]x[\/latex] does determine a unique value for [latex]y,[\/latex] and there are mathematical procedures by which [latex]y[\/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[\/latex] as a function of [latex]x,[\/latex] even though the formula cannot be written explicitly.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165137648450\" class=\"bc-section section\">\n<h4>Evaluating a Function Given in Tabular Form<\/h4>\n<p id=\"fs-id1165135186424\">As we saw previously, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.<\/p>\n<p id=\"fs-id1165135186427\">The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See <a class=\"autogenerated-content\" href=\"#Table_01_01_10\">Table 10<\/a>.<a class=\"footnote\" title=\"http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590. Accessed 3\/24\/2014.\" id=\"return-footnote-26-2\" href=\"#footnote-26-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<table id=\"Table_01_01_10\" style=\"height: 158px; width: 200px;\" summary=\"Six rows and two columns. The first column is labeled, \u201cpet\u201d, and the second column is labeled, \u201cmemory span in hours\u201d. Reading the rows as ordered pairs, we have: (puppy, 0.008), (adult dog, 0.083), (cat, 16), (goldfish, 2100), and (beta fish, 3600).\">\n<caption><strong>Table 10<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Pet<\/td>\n<td class=\"border\" style=\"width: 50%; text-align: center;\">Memory span in hours<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Puppy<\/td>\n<td class=\"border\" style=\"width: 50%; text-align: center;\">0.008<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Adult dog<\/td>\n<td class=\"border\" style=\"width: 50%; text-align: center;\">0.083<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Cat<\/td>\n<td class=\"border\" style=\"width: 50%; text-align: center;\">16<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Goldfish<\/td>\n<td class=\"border\" style=\"width: 50%; text-align: center;\">2160<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 113.75px; text-align: center;\">Beta fish<\/td>\n<td class=\"border\" style=\"width: 50%; text-align: center;\">3600<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137584852\">At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P.[\/latex] The <span class=\"no-emphasis\">domain<\/span> of the function is the type of pet and the range is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of \u201cgoldfish.\u201d We would write [latex]P\\left(\\text{goldfish}\\right)=2160.[\/latex] Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[\/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.<\/p>\n<div id=\"fs-id1165137838337\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137870786\"><strong>Given a function represented by a table, identify specific output and input values. <\/strong><\/p>\n<ol id=\"fs-id1165137870791\" type=\"1\">\n<li>Find the given input in the row (or column) of input values.<\/li>\n<li>Identify the corresponding output value paired with that input value.<\/li>\n<li>Find the given output values in the row (or column) of output values, noting every time that output value appears.<\/li>\n<li>Identify the input value(s) corresponding to the given output value.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_01_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137619419\">\n<div id=\"fs-id1165137619421\">\n<h3>Example 11:\u00a0 Evaluating and Solving a Tabular Function<\/h3>\n<p id=\"fs-id1165133356033\">Using <a class=\"autogenerated-content\" href=\"#Table_01_01_11\">Table 11<\/a>,<\/p>\n<ol id=\"fs-id1165137653327\" type=\"a\">\n<li>Evaluate [latex]g\\left(3\\right).[\/latex]<\/li>\n<li>Solve [latex]g\\left(n\\right)=6.[\/latex]<\/li>\n<\/ol>\n<table id=\"Table_01_01_11\" summary=\"Two rows and six columns. The first row is labeled, \u201cn\u201d and the second row is labeled, \u201cg(n)\u201d. Reading the columns as ordered pairs, we have: (1, 8), (2, 6), (3, 7), (4, 6) , and (5, 8).\">\n<caption><strong>Table 11<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 134.775px;\"><strong>[latex]n[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">4<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1875px; text-align: center;\">5<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 134.775px;\"><strong>[latex]g\\left(n\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">8<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1625px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"height: 12px; width: 24.1875px; text-align: center;\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137748378\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137748378\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137748378\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137725812\" type=\"a\">\n<li>Evaluating [latex]g\\left(3\\right)[\/latex] means determining the output value of the function [latex]g[\/latex] for the input value of [latex]n=3.[\/latex] The table output value corresponding to [latex]n=3[\/latex] is 7, so [latex]g\\left(3\\right)=7.[\/latex]<\/li>\n<li>Solving [latex]g\\left(n\\right)=6[\/latex] means identifying the input values, [latex]n,[\/latex] that produce an output value of 6. <a class=\"autogenerated-content\" href=\"#Table_01_01_12\">Table 11<\/a> shows two solutions: [latex]2[\/latex] and [latex]4.[\/latex]\u00a0 When we input 2 into the function [latex]g,[\/latex] our output is 6. When we input 4 into the function [latex]g,[\/latex] our output is also 6.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137584384\" class=\"precalculus tryit\">\n<h3>Try it #7<\/h3>\n<div id=\"ti_01_01_06\">\n<div id=\"fs-id1165137557816\">\n<p id=\"fs-id1165137557817\">Using <a class=\"autogenerated-content\" href=\"#Table_01_01_12\">Table 11<\/a>, evaluate [latex]g\\left(1\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137423936\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137423936\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137423936\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137423937\">[latex]g\\left(1\\right)=8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135696152\" class=\"bc-section section\">\n<h4>Finding Function Values from a Graph<\/h4>\n<p id=\"fs-id1165137779152\">Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\n<div id=\"Example_01_01_12\" class=\"textbox examples\">\n<div id=\"fs-id1165134212105\">\n<div id=\"fs-id1165134212107\">\n<h3>Example 12:\u00a0 Reading Function Values from a Graph<\/h3>\n<p id=\"fs-id1165137469316\">Given the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_01_007\">Figure 7<\/a>,<\/p>\n<ol id=\"fs-id1165137604039\" type=\"a\">\n<li>Evaluate [latex]f\\left(2\\right).[\/latex]<\/li>\n<li>Solve [latex]f\\left(x\\right)=4.[\/latex]<\/li>\n<\/ol>\n<div id=\"attachment_3029\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3029\" class=\"size-full wp-image-3029\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142815\/985e7ba8c4bba18084e0450b429d6a0b53dfd707-1.jpeg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/><\/p>\n<p id=\"caption-attachment-3029\" class=\"wp-caption-text\"><strong>Figure 7<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_007\" class=\"small\">\n<div class=\"mceTemp\"><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137849160\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137849160\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137849160\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137871522\" type=\"a\">\n<li>To evaluate [latex]f\\left(2\\right),[\/latex] locate the point on the curve where [latex]x=2,[\/latex] then read the <em>y<\/em>-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right),[\/latex] so [latex]f\\left(2\\right)=1.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_01_008\">Figure 8<\/a>\n<div id=\"attachment_3030\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3030\" class=\"size-full wp-image-3030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14142949\/2d4c4269758c5a422deed68865384d7b6774b924-1.jpeg\" alt=\"The same parabola as Figure 7, now with the point (2,1) labeled and the equation f(2)=1\" width=\"487\" height=\"445\" \/><\/p>\n<p id=\"caption-attachment-3030\" class=\"wp-caption-text\"><strong>Figure 8<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_008\" class=\"small\"><\/div>\n<\/li>\n<li>To solve [latex]f\\left(x\\right)=4,[\/latex] we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4,[\/latex] we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right).[\/latex] These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]-1[\/latex] or [latex]3.[\/latex] This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4,[\/latex] or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_01_009\">Figure 9<\/a>.\n<div id=\"attachment_3031\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3031\" class=\"size-full wp-image-3031\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143208\/4a94983cb7a344c2cba4c4a8bb3ed7e9f1f0b084-1.jpeg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/><\/p>\n<p id=\"caption-attachment-3031\" class=\"wp-caption-text\"><strong>Figure 9<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_009\" class=\"small\">\n<div class=\"mceTemp\"><\/div>\n<p><span id=\"fs-id1165137469773\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135149263\" class=\"precalculus tryit\">\n<h3>Try it #8<\/h3>\n<div id=\"ti_01_01_05\">\n<div id=\"fs-id1165137695207\">\n<p id=\"fs-id1165137695208\">Using <a class=\"autogenerated-content\" href=\"#Figure_01_01_007\">Figure 7<\/a>, solve [latex]f\\left(x\\right)=1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137598286\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137598286\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137598286\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137598287\">[latex]x=0[\/latex] or [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135422920\" class=\"bc-section section\">\n<h3>Determining Whether a Function is One-to-One<\/h3>\n<p id=\"fs-id1165135678633\">Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in <a href=\"#figure10_1_add\">Figure 10<\/a>, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.<a id=\"figure10_1_add\"><\/a><\/p>\n<div id=\"attachment_1600\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1600\" class=\"size-medium wp-image-1600\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/18121430\/Bull-Market-Graph-Abramson-ch1-300x164.png\" alt=\"\" width=\"300\" height=\"164\" \/><\/p>\n<p id=\"caption-attachment-1600\" class=\"wp-caption-text\">Figure 10<\/p>\n<\/div>\n<p id=\"fs-id1165135245630\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in <a class=\"autogenerated-content\" href=\"#Table_01_01_13\">Table 12<\/a>.<\/p>\n<table id=\"Table_01_01_13\" style=\"height: 132px; width: 200px;\" summary=\"Two columns and five rows. The first column is labeled, \u201cLetter Grade\u201d, and the second column is labeled, \u201cGrade point average\u201d. Reading the rows as ordered pairs, we have: (A, 4.0), (B, 3.0), (C, 2.0), and (D, 1.0).\">\n<caption><strong>Table 12<\/strong><\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<td class=\"border\">Letter grade<\/td>\n<td class=\"border\">Grade point average<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\">A<\/td>\n<td class=\"border\">4.0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">B<\/td>\n<td class=\"border\">3.0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">C<\/td>\n<td class=\"border\">2.0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">D<\/td>\n<td class=\"border\">1.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n<p id=\"fs-id1165137628999\">To visualize this concept, let\u2019s look again at the two simple functions sketched in <a class=\"autogenerated-content\" href=\"#Figure_01_01_001\">Figure 1<\/a><strong>(a) <\/strong>and <a class=\"autogenerated-content\" href=\"#Figure_01_01_001\">Figure 1<\/a><strong>(b)<\/strong>. The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n.[\/latex] The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p>A <em><strong>one-to-one function<\/strong><\/em> is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<div id=\"Example_01_01_13\" class=\"textbox examples\">\n<div id=\"fs-id1165137755749\">\n<div id=\"fs-id1165137755752\">\n<h3>Example 13:\u00a0 Determining Whether a Relationship Is a One-to-One Function<\/h3>\n<p id=\"fs-id1165137892387\">Is the area of a circle a function of its radius? If yes, is the function one-to-one?<\/p>\n<\/div>\n<div id=\"fs-id1165137892391\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137892391\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137892391\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134148470\">A circle of radius [latex]r[\/latex] has a unique area measure given by [latex]A=\\pi {r}^{2},[\/latex] so for any input, [latex]r,[\/latex] there is only one output, [latex]A.[\/latex] The area is a function of radius [latex]r.[\/latex]<\/p>\n<p id=\"fs-id1165137892326\">If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[\/latex] is given by the formula [latex]A=\\pi {r}^{2}.[\/latex] Because areas and radii are positive numbers, there is exactly one solution: [latex]\\sqrt{\\frac{A}{\\pi }}.[\/latex] So the area of a circle is a one-to-one function of the circle\u2019s radius.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137579363\" class=\"precalculus tryit\">\n<h3>Try it #9<\/h3>\n<div id=\"ti_01_01_10\">\n<div id=\"fs-id1165134079641\">\n<ol>\n<li>Is a balance a function of the bank account number?<\/li>\n<li>Is a bank account number a function of the balance?<\/li>\n<li>Is a balance a one-to-one function of the bank account number?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137456018\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137456018\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137456018\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li id=\"fs-id1165137592691\">Yes, because each bank account has a single balance at any given time;<\/li>\n<li>No, because several bank account numbers may have the same balance;<\/li>\n<li>No, because the same output may correspond to more than one input.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1327356\" class=\"precalculus tryit\">\n<h3>Try it #10<\/h3>\n<div id=\"ti_01_01_11\">\n<div id=\"fs-id1165137737576\">\n<p id=\"eip-id1166399990268\">Evaluate the following:<\/p>\n<ol id=\"fs-id1165135160248\" type=\"a\">\n<li>If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade? Explain.<\/li>\n<li>If so, is the function one-to-one? Explain.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137655289\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137655289\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137655289\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137655291\" type=\"a\">\n<li>Yes, the letter grade is a function of percent grade. Each input or percent grade is mapped to exactly one letter grade.<\/li>\n<li>No, it is not one-to-one. Each letter grade must be associated with more than one input.\u00a0 There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135435781\" class=\"bc-section section\">\n<h3>Using the Vertical Line Test<\/h3>\n<p id=\"fs-id1165135435786\">As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\n<p id=\"fs-id1165137637786\">Very often graphs name the input value [latex]x[\/latex] and the output value [latex]y,[\/latex] and we say [latex]y[\/latex] is a function of [latex]x,[\/latex] or [latex]y=f\\left(x\\right)[\/latex] when the function is named [latex]f.[\/latex] The graph of the function is the set of all points [latex]\\left(x,y\\right)[\/latex] in the plane that satisfies the equation [latex]y=f\\left(x\\right).[\/latex] If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em>x<\/em>-coordinate of each point is an input value and the <em>y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_01_011\">Figure 11<\/a> tell us that [latex]f\\left(0\\right)=2[\/latex] and [latex]f\\left(6\\right)=1.[\/latex] However, the set of all points [latex]\\left(x,y\\right)[\/latex] satisfying [latex]y=f\\left(x\\right)[\/latex] is a curve. The curve shown includes [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex] because the curve passes through those points.<\/p>\n<div id=\"attachment_3032\" style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3032\" class=\"size-full wp-image-3032\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143447\/659b097af5e55cf05e8eba3f47d782bff5c352a7-2.jpeg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" \/><\/p>\n<p id=\"caption-attachment-3032\" class=\"wp-caption-text\"><strong>Figure 11<\/strong><\/p>\n<\/div>\n<p id=\"fs-id1165137737620\">The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because a function can have only one output value for each input value. See <a class=\"autogenerated-content\" href=\"#Figure_01_01_012\">Figure 12<\/a>.<\/p>\n<div id=\"attachment_3033\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3033\" class=\"size-full wp-image-3033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143547\/73a7944f1a6253e4cc8726cc09e92801c34c9d76.jpeg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/><\/p>\n<p id=\"caption-attachment-3033\" class=\"wp-caption-text\"><strong>Figure 12<\/strong><\/p>\n<\/div>\n<p>The second and third graphs in Figure 12 are not functions because the input value represented by the dotted line has two output values in each case.\u00a0 The two output values are the [latex]y[\/latex] values where each dotted line intersects the solid line.\u00a0 This contradicts the definition of a function. Remember, a function is a rule where each input is mapped to\u00a0<strong>exactly\u00a0<\/strong>one output.<\/p>\n<div id=\"fs-id1165135460884\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137452182\"><strong>Given a graph, use the vertical line test to determine if the graph represents a function. <\/strong><\/p>\n<ol id=\"fs-id1165133277614\" type=\"1\">\n<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, determine that the graph does not represent a function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_01_14\" class=\"textbox examples\">\n<div id=\"fs-id1165134541166\">\n<div id=\"fs-id1165137571591\">\n<h3>Example 14:\u00a0 Applying the Vertical Line Test<\/h3>\n<p id=\"fs-id1165137761111\">Which of the graphs in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a> represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<\/p>\n<div id=\"attachment_3034\" style=\"width: 985px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3034\" class=\"size-full wp-image-3034\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143659\/9102dcc3fb5ba81653f83c45e2a35606c7e74b44.jpeg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<p id=\"caption-attachment-3034\" class=\"wp-caption-text\"><strong>Figure 13<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190052\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135190052\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135190052\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137629350\">If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a>. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at\u00a0<em>x<\/em>-values between -3 and 3, a vertical line would intersect the graph at more than one point, as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_016\">Figure 14<\/a>.\u00a0 This indicates that each of these inputs gets mapped to two different outputs.\u00a0 This contradicts the definition of a function, since we know a function maps each input to exactly one output.<\/p>\n<div id=\"attachment_3035\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3035\" class=\"size-full wp-image-3035\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143748\/4f55692043ccebf4409fbd35370e2d0aa41230cd.jpeg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/><\/p>\n<p id=\"caption-attachment-3035\" class=\"wp-caption-text\"><strong>Figure 14<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_016\" class=\"small\">\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134544969\" class=\"precalculus tryit\">\n<h3>Try it #11<\/h3>\n<div id=\"ti_01_01_04\">\n<div id=\"fs-id1165135600805\">\n<p id=\"fs-id1165135210137\">Does the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_01_017\">Figure 15<\/a> represent a function?<\/p>\n<div id=\"attachment_3036\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3036\" class=\"size-full wp-image-3036\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14143851\/6d3f6d402260a9c5f6e1312c594bd7b89db74e02.jpeg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/><\/p>\n<p id=\"caption-attachment-3036\" class=\"wp-caption-text\"><strong>Figure 15<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"small\"><\/div>\n<div id=\"ti_01_01_04\">\n<div id=\"fs-id1165134258608\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134258608\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134258608\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134258609\">Yes.\u00a0 Any vertical line will only pass through the graph once.\u00a0 This confirms the definition of one-to-one, since each output corresponds to only one input.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137610952\" class=\"bc-section section\">\n<h3>Using the Horizontal Line Test<\/h3>\n<p id=\"fs-id1165137871503\">Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.\u00a0 Each intersection along the horizontal line represents an x-value with the same output which contradicts the definition of one-to-one which states that each\u00a0 output value must be unique for the function to be one-to-one.<\/p>\n<div id=\"fs-id1165137736232\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165133437255\"><strong>Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function. <\/strong><\/p>\n<ol id=\"fs-id1165137611853\" type=\"1\">\n<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, determine that the function is not one-to-one.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_01_15\" class=\"textbox examples\">\n<div id=\"fs-id1165134389035\">\n<div id=\"fs-id1165134342668\">\n<h3>Example 15:\u00a0 Applying the Horizontal Line Test<\/h3>\n<p id=\"fs-id1165135434808\">Consider the functions shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13 <\/a><strong>(a)<\/strong> and <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a><strong>(b)<\/strong>. Are either of the functions one-to-one?<\/p>\n<\/div>\n<div id=\"fs-id1165135521259\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135521259\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135521259\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135185190\">The function in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a><strong>(a) <\/strong>is not one-to-one. The horizontal line shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_010\">Figure 16<\/a> intersects the graph of the function at two points.\u00a0 These two points have the same output value but different input values.\u00a0 Remember that the definition of a one-to-one function is that each output value corresponds to exactly one input value.\u00a0 Therefore, when a horizontal line intersects a graph at more than one point, we have a contradiction to this definition, and the function cannot be a one-to-one function.<\/p>\n<div id=\"attachment_3037\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3037\" class=\"wp-image-3037 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14144007\/140fa74177d512d2a0de3741db0fb44d27334d60.jpeg\" alt=\"The horizontal line drawn intersects the polynomial two times.\" width=\"487\" height=\"445\" \/><\/p>\n<p id=\"caption-attachment-3037\" class=\"wp-caption-text\"><strong>Figure 16<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_01_010\" class=\"small\"><\/div>\n<p id=\"fs-id1165135151243\">The function in <a class=\"autogenerated-content\" href=\"#Figure_01_01_013\">Figure 13<\/a><strong>(b)<\/strong> is one-to-one. Any horizontal line will intersect a diagonal line at most once.\u00a0 This means that each output is associated with only one input value, satisfying the definition of a one-to-one function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135252051\" class=\"precalculus tryit\">\n<h3>Try it #12<\/h3>\n<div id=\"ti_01_01_12\">\n<div id=\"fs-id1165137749742\">\n<p id=\"fs-id1165137749744\">Is the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_01_01_016\">Figure 13<\/a> <strong>(c)<\/strong> one-to-one?<\/p>\n<\/div>\n<div id=\"fs-id1165135255384\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135255384\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135255384\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135255385\">No, because it does not pass the horizontal line test.\u00a0 We can see that there are horizontal lines that would intersect the graph in more than one point, indicating that an output value has more than one input value that corresponds to it.\u00a0 This contradicts the definition of one-to-one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545919\" class=\"bc-section section\">\n<h3>Identifying Basic Toolkit Functions<\/h3>\n<p id=\"fs-id1165137698132\">In this text, we will be exploring functions including the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our \u201ctoolkit functions,\u201d which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[\/latex] as the input variable and [latex]y=f\\left(x\\right)[\/latex] as the output variable.<\/p>\n<p id=\"fs-id1165135591070\">We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in <a class=\"autogenerated-content\" href=\"#Table_01_01_14\">Table 13<\/a>.<\/p>\n<table id=\"Table_01_01_14\" style=\"height: 2048px;\" summary=\"The title is \u201cToolkit functions\u201d. There are three columns and ten rows. The first column is labeled, \u201cname\u201d, the second column is labeled, \u201cfunction\u201d, and the third column is labeled graph which contains pictures of the functions. The constant function is f(x) = c where c is the constant; the identity function is f(x) = x; the absolute function is f(x)=|x|; the quadratic function is f(x) = x^2; the cubic function is f(x)=x^3; the reciprocal function is f(x)=1\/x; the reciprocal squared function is f(x)=1\/x^2; the square root function is f(x)=sqrt(x); the cube root function is f(x) = x^(1\/3).\">\n<caption>Table 13<\/caption>\n<thead>\n<tr style=\"height: 74px;\">\n<td class=\"border\" style=\"text-align: center; width: 1006.5px; height: 74px;\" colspan=\"3\">\n<h1>Toolkit Functions<\/h1>\n<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 110.5px;\">Name<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">Function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 517.5px;\">Graph<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 320px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Constant<\/td>\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=c,[\/latex] where [latex]c[\/latex] is a constant<\/td>\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137643159\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3067\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151148\/c166f7a1121575944ac02d2fc5c70770d0e963e6.jpeg\" alt=\"Graph of a constant function.\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 320px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Identity<\/td>\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137811013\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3068 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151218\/76a34ffd5fe5dc36af950679c61c54daeaf97c06.jpeg\" alt=\"Graph of a straight line, slope of 1.\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 320px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Absolute value<\/td>\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165135195221\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3069\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151319\/ec22554e96a17ac92c66de3864e19c40780587dd.jpeg\" alt=\"Graph of an absolute value\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 320px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Quadratic<\/td>\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137501903\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3070\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151357\/0f74efa823699938eb96f0ab6f767f6a545b525a.jpeg\" alt=\"Graph of a parabola\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 320px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Cubic<\/td>\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165137722123\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3071\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151429\/45dd3744b83abcb7676e82f77770ce461218c2e7.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 320px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 320px; width: 110.5px;\">Reciprocal<\/td>\n<td class=\"border\" style=\"height: 320px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 320px; width: 517.5px;\"><span id=\"fs-id1165134544980\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3072\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151453\/0d2bcdc4558ed0b1afdbadc6e5b1c72b22985cdf.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 14px; width: 110.5px;\">Reciprocal squared<\/td>\n<td class=\"border\" style=\"height: 14px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 14px; width: 517.5px;\"><span id=\"fs-id1165137647610\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3073\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151527\/a33ee546ef2cbaa4a5ee7ee57c20463ec6801fff.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 14px; width: 110.5px;\">Square root<\/td>\n<td class=\"border\" style=\"height: 14px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 14px; width: 517.5px;\"><span id=\"fs-id1165137863670\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3074\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151553\/de470da28ca919df70b52178fd96577e7040eb51.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\" valign=\"top\">\n<td class=\"border\" style=\"height: 14px; width: 110.5px;\">Cube root<\/td>\n<td class=\"border\" style=\"height: 14px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 14px; width: 517.5px;\"><span id=\"fs-id1165137838612\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3075\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151627\/b7449e93ba1d9ebee82fbea8aea5676948ce9af9.jpeg\" alt=\"\" width=\"517\" height=\"319\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165134042311\" class=\"precalculus media\">\n<p id=\"fs-id1165135549046\">Access the following online resources for additional instruction and practice with functions.<\/p>\n<ul id=\"eip-id1165137846437\">\n<li><a href=\"http:\/\/openstax.org\/l\/relationfunction\">Determine if a Relation is a Function<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine if a Relation is a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zT69oxcMhPw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"eip-id1165137846437\">\n<li><a href=\"http:\/\/openstax.org\/l\/vertlinetest\">Vertical Line Test<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"The Vertical Line Test\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/gO5WN9g1fJo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"eip-id1165137846437\">\n<li><a href=\"http:\/\/openstax.org\/l\/introtofunction\">Introduction to Functions<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Introduction to Functions - Part 2\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/sW9-zBeQpCU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"eip-id1165137846437\">\n<li><a href=\"http:\/\/openstax.org\/l\/vertlinegraph\">Vertical Line Test on Graph<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5Z8DaZPJLKY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"eip-id1165137846437\">\n<li><a href=\"http:\/\/openstax.org\/l\/onetoone\">One-to-one Functions<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Determine if a Relation Given as a Table is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QFOJmevha_Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"eip-id1165137846437\">\n<li><a href=\"http:\/\/openstax.org\/l\/graphonetoone\">Graphs as One-to-one Functions<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex 1:  Determine if the Graph of a Relation is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tbSGdcSN8RE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203679\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134393730\" style=\"height: 211px;\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Constant function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=c,[\/latex] where [latex]c[\/latex] is a constant<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Identity function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Absolute value function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Quadratic function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Cubic function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Reciprocal function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Reciprocal squared function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 159.5px;\">Square root function<\/td>\n<td class=\"border\" style=\"height: 12px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td class=\"border\" style=\"height: 44px; width: 159.5px;\">Cube root function<\/td>\n<td class=\"border\" style=\"height: 44px; width: 353.5px;\">[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 71px;\">\n<td class=\"border\" style=\"height: 71px; width: 159.5px;\">Difference Quotient<\/td>\n<td class=\"border\" style=\"height: 71px; width: 353.5px;\">[latex]\\frac{f(a+h)-f(a)}{h}\\\\[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137692068\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137851183\">\n<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\n<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right).[\/latex]<\/li>\n<li>In tabular form, a function can be represented by rows or columns that relate to input and output values.<\/li>\n<li>To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.<\/li>\n<li>To solve for a specific function value, we determine the input values that yield the specific output value.<\/li>\n<li>An algebraic form of a function can be written from an equation.<\/li>\n<li>Input and output values of a function can be identified from a table.<\/li>\n<li>Relating input values to output values on a graph is another way to evaluate a function.<\/li>\n<li>A function is one-to-one if each output value corresponds to only one input value.<\/li>\n<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\n<li>The graph of a one-to-one function passes the horizontal line test.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137758543\">\n<dt>dependent variable<\/dt>\n<dd id=\"fs-id1165137758548\">an output variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137758552\">\n<dt>domain<\/dt>\n<dd id=\"fs-id1165137932576\">the set of all possible input values for a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932580\">\n<dt>function<\/dt>\n<dd id=\"fs-id1165137932585\">a relation in which each input value yields a unique output value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932588\">\n<dt>horizontal line test<\/dt>\n<dd id=\"fs-id1165134149777\">a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134149782\">\n<dt>independent variable<\/dt>\n<dd id=\"fs-id1165134149787\">an input variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511353\">\n<dt>input<\/dt>\n<dd id=\"fs-id1165135511359\">each object or value in a domain that relates to another object or value by a relationship known as a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511364\">\n<dt>one-to-one function<\/dt>\n<dd id=\"fs-id1165135511369\">a function for which each value of the output is associated with a unique input value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508564\">\n<dt>output<\/dt>\n<dd id=\"fs-id1165135508569\">each object or value in the range that is produced when an input value is entered into a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508573\">\n<dt>range<\/dt>\n<dd id=\"fs-id1165135315529\">the set of output values that result from the input values in a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315533\">\n<dt>relation<\/dt>\n<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315542\">\n<dt>vertical line test<\/dt>\n<dd id=\"fs-id1165134186374\">a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-26\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li> Functions and Function Notation. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:EfTqzMNI@13\/Functions-and-Function-Notation\">https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:EfTqzMNI@13\/Functions-and-Function-Notation<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 1. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-26-1\"><a href=\"http:\/\/www.baseball-almanac.com\/legendary\/lisn100.shtml\">http:\/\/www.baseball-almanac.com\/legendary\/lisn100.shtml<\/a>. Accessed 3\/24\/2014 <a href=\"#return-footnote-26-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-26-2\"><a href=\"http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590\">http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590<\/a>. Accessed 3\/24\/2014. <a href=\"#return-footnote-26-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":311,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\" Functions and Function Notation\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:EfTqzMNI@13\/Functions-and-Function-Notation\",\"project\":\"Essential Precalcus, Part 1\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-26","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/26","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":62,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/26\/revisions"}],"predecessor-version":[{"id":3291,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/26\/revisions\/3291"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/26\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=26"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=26"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=26"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=26"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}