{"id":1780,"date":"2023-10-12T00:32:09","date_gmt":"2023-10-12T00:32:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-characteristics-of-functions-and-their-graphs\/"},"modified":"2026-01-15T19:02:28","modified_gmt":"2026-01-15T19:02:28","slug":"introduction-characteristics-of-functions-and-their-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-characteristics-of-functions-and-their-graphs\/","title":{"raw":"Functions","rendered":"Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul class=\"ul1\">\r\n \t<li class=\"li2\"><span class=\"s1\">Determine whether a relation represents a function.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Find function values.<\/span><\/li>\r\n \t<li>Use functional notation.<\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Use the vertical line test to identify functions.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Graph parent functions.<\/span><\/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<h2>Characteristics of Functions<\/h2>\r\nA <strong>relation<\/strong> is a set of ordered pairs. The set of the first components of each <strong>ordered pair<\/strong> is called the <strong>domain <\/strong>of the relation\u00a0and the set of the second components of each ordered pair is called the <strong>range\u00a0<\/strong>of the relation. 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 the first.\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(1,2\\right),\\left(2,4\\right),\\left(3,6\\right),\\left(4,8\\right),\\left(5,10\\right)\\right\\}[\/latex]<\/p>\r\nThe domain is [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0The range is [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].\r\n\r\nNote the values in the domain are also known as an <strong>input<\/strong> values, or values of the\u00a0<strong>independent variable<\/strong>, and are often labeled with the lowercase letter [latex]x[\/latex]. Values in the range are also known as an <strong>output<\/strong> values, or values of the\u00a0<strong>dependent variable<\/strong>, and are often labeled with the lowercase letter [latex]y[\/latex].\r\n\r\nA <strong>function<\/strong> [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 used more than once. For our example that relates the first five <strong>natural numbers<\/strong> to numbers double their values, this relation is a function because each element in the domain, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex], is paired with exactly one element in the range, [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].\r\n\r\nNow let\u2019s consider the set of ordered pairs that relates the terms \"even\" and \"odd\" to the first five natural numbers. It would appear as\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(\\text{odd},1\\right),\\left(\\text{even},2\\right),\\left(\\text{odd},3\\right),\\left(\\text{even},4\\right),\\left(\\text{odd},5\\right)\\right\\}[\/latex]<\/p>\r\nNotice that each element in the domain, [latex]\\left\\{\\text{even,}\\text{odd}\\right\\}[\/latex]\u00a0is <em>not<\/em> paired with exactly one element in the range, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0For example, the term \"odd\" corresponds to three values from the domain, [latex]\\left\\{1,3,5\\right\\}[\/latex]\u00a0and the term \"even\" corresponds to two values from the range, [latex]\\left\\{2,4\\right\\}[\/latex].\u00a0This violates the definition of a function, so this relation is not a function.\r\n\r\nThis image compares relations that are functions and not functions.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190946\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/> (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 class=\"textbox\">\r\n<h3>A General Note: FunctionS<\/h3>\r\nA <strong>function<\/strong> is a relation in which each possible input value leads to exactly one output value. We say \"the output is a function of the input.\"\r\n\r\nThe <strong>input<\/strong> values make up the <strong>domain<\/strong>, and the <strong>output<\/strong> values make up the <strong>range<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3><strong>How To: Given a relationship between two quantities, determine whether the relationship is a function.<\/strong><\/h3>\r\n<ol>\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, the relationship is a function. If any input value leads to two or more outputs, the relationship is not a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining If Menu Price Lists Are Functions<\/h3>\r\nThe coffee shop menu consists of items and their prices.\r\n<ol>\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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190949\/CNX_Precalc_Figure_01_01_0042.jpg\" 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\" \/>\r\n[reveal-answer q=\"507796\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507796\"]\r\n<ol>\r\n \t<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190951\/CNX_Precalc_Figure_01_01_0272.jpg\" 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\" \/>Each item on the menu has only one price, so the price is a function of the item.<\/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.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190954\/CNX_Precalc_Figure_01_01_0282.jpg\" alt=\"Association of the prices to the donuts.\" width=\"731\" height=\"241\" \/>Therefore, the item is a not a function of price.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining If Class Grade Rules Are Functions<\/h3>\r\nIn 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? The table below shows a possible rule for assigning grade points.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Percent Grade<\/th>\r\n<td>0\u201356<\/td>\r\n<td>57\u201361<\/td>\r\n<td>62\u201366<\/td>\r\n<td>67\u201371<\/td>\r\n<td>72\u201377<\/td>\r\n<td>78\u201386<\/td>\r\n<td>87\u201391<\/td>\r\n<td>92\u2013100<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>Grade Point Average<\/th>\r\n<td>0.0<\/td>\r\n<td>1.0<\/td>\r\n<td>1.5<\/td>\r\n<td>2.0<\/td>\r\n<td>2.5<\/td>\r\n<td>3.0<\/td>\r\n<td>3.5<\/td>\r\n<td>4.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"813427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"813427\"]\r\n\r\nFor 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.\r\n\r\nIn 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/zT69oxcMhPw\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe table below\u00a0lists the five greatest baseball players of all time in order of rank.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Player<\/th>\r\n<th>Rank<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Babe Ruth<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Willie Mays<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Ty Cobb<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Walter Johnson<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Hank Aaron<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\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[reveal-answer q=\"112010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"112010\"]\r\n<ol>\r\n \t<li>yes<\/li>\r\n \t<li>yes.\u00a0(Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=111625&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Using Function Notation<\/h3>\r\nOnce 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 <strong>function notation<\/strong> is one representation that makes it easier to work\u00a0with functions.\r\n\r\nTo represent \"height is a function of age,\" we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age. The letters [latex]f,g[\/latex], and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,y[\/latex], and [latex]z[\/latex] to represent numbers and [latex]A,B[\/latex],\u00a0and [latex]C[\/latex] to represent sets.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;h\\text{ is }f\\text{ of }a &amp;&amp;\\text{We name the function }f;\\text{ height is a function of age}. \\\\ &amp;h=f\\left(a\\right) &amp;&amp;\\text{We use parentheses to indicate the function input}\\text{. } \\\\ &amp;f\\left(a\\right) &amp;&amp;\\text{We name the function }f;\\text{ the expression is read as }\"f\\text{ of }a\". \\end{align}[\/latex]<\/p>\r\nRemember, we can use any letter to name the function; we can use the notation [latex]h\\left(a\\right)[\/latex]\u00a0 to show that [latex]h[\/latex] depends on [latex]a[\/latex]. The input value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get an output value. The parentheses indicate that age is input into the function; they do not indicate multiplication.\r\n\r\nWe can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means \"first add [latex]a[\/latex]\u00a0and [latex]b[\/latex], and the result is the input for the function [latex]f[\/latex].\" We must perform the operations in this order to obtain the correct result.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Function Notation<\/h3>\r\nThe notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as [latex]\"y[\/latex] is a function of [latex]x.\"[\/latex] The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex], or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Function Notation for Days in a Month<\/h3>\r\nUse function notation to represent a function whose input is the name of a month and output is the number of days in that month in a non-leap year.\r\n\r\n[reveal-answer q=\"349740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"349740\"]\r\n\r\nThe 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]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a \"rule\" that associates a specific number (the output) with each input.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190956\/CNX_Precalc_Figure_01_01_0052.jpg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" \/>\r\n\r\nFor example, [latex]f\\left(\\text{April}\\right)=30[\/latex], because April has 30 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).\r\n<h4>Analysis of the Solution<\/h4>\r\nWe must restrict the function to non-leap years. Otherwise February would have 2 outputs and this would not be a function. Also 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 as inputs and outputs.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Interpreting Function Notation<\/h3>\r\nA 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?\r\n\r\n[reveal-answer q=\"299999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"299999\"]\r\n\r\nWhen 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2510&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<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 \"<em>y<\/em> is a function of <em>x<\/em>?\"<\/strong>\r\n\r\n<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>\r\n\r\n<\/div>\r\n<h2>Representing Functions Using Tables<\/h2>\r\nA common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values.\u00a0In 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.\r\n\r\nThe table below 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.\r\n<table summary=\"Two rows and thirteen columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Month number, [latex]m[\/latex] (input)<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<td>9<\/td>\r\n<td>10<\/td>\r\n<td>11<\/td>\r\n<td>12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Days in month, [latex]D[\/latex] (output)<\/strong><\/td>\r\n<td>31<\/td>\r\n<td>28<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<td>30<\/td>\r\n<td>31<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table below\u00a0defines 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[\/latex].\r\n<table summary=\"Two rows and six columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]n[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]Q[\/latex]<\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table below 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.\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Age in years, [latex]\\text{ }a\\text{ }[\/latex] (input)<\/strong><\/td>\r\n<td>5<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<td>9<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Height in inches, [latex]\\text{ }h\\text{ }[\/latex] (output)<\/strong><\/td>\r\n<td>40<\/td>\r\n<td>42<\/td>\r\n<td>44<\/td>\r\n<td>47<\/td>\r\n<td>50<\/td>\r\n<td>52<\/td>\r\n<td>54<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3><strong>How To: Given a table of input and output values, determine whether the table represents a function.\r\n<\/strong><\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Identifying Tables that Represent Functions<\/h3>\r\nWhich table, A, B, or C, represents a function (if any)?\r\n<table summary=\"Four rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Table A<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Input<\/th>\r\n<th>Output<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table summary=\"Four rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Table B<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Input<\/th>\r\n<th>Output<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\u20133<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table summary=\"Four rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Table C<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Input<\/th>\r\n<th>Output<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"979211\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"979211\"]\r\n\r\na)\u00a0and b)\u00a0define functions. In both, each input value corresponds to exactly one output value. c)\u00a0does not define a function because the input value of 5 corresponds to two different output values.\r\n\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)\u00a0can be represented by writing\r\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=1,f\\left(5\\right)=3,\\text{and }f\\left(8\\right)=6[\/latex]<\/p>\r\nSimilarly, the statements\u00a0[latex]g\\left(-3\\right)=5,g\\left(0\\right)=1,\\text{and }g\\left(4\\right)=5[\/latex]\u00a0represent the function in b).\r\n\r\nc)\u00a0cannot be expressed in a similar way because it does not represent a function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen we know an input value and want to determine the corresponding output value for a function, we <em>evaluate<\/em> the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.\r\n\r\nWhen 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 <em>solve<\/em> for the input. Solving can produce more than one solution because different input values can produce the same output value.\r\n\r\n\r\n<\/div>\r\n<h2>Evaluating and Solving Functions<\/h2>\r\nWhen 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.\r\n<div class=\"textbox\">\r\n<h3><strong>How To: EVALUATE A FUNCTION Given ITS FORMula.\r\n<\/strong><\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Evaluating Functions<\/h3>\r\nGiven the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], evaluate [latex]h\\left(4\\right)[\/latex].\r\n\r\n[reveal-answer q=\"768180\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"768180\"]\r\n<p style=\"text-align: left;\">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 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\nTherefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/Ehkzu5Uv7O0\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Functions at Specific Values<\/h3>\r\nFor the function, [latex]f\\left(x\\right)={x}^{2}+3x - 4[\/latex], evaluate each of the following.\r\n<ol>\r\n \t<li>[latex]f\\left(2\\right)[\/latex]<\/li>\r\n \t<li>[latex]f(a)[\/latex]<\/li>\r\n \t<li>[latex]f(a+h)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"645951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"645951\"]\r\n\r\nReplace the [latex]x[\/latex]\u00a0in the function with each specified value.\r\n<ol>\r\n \t<li>Because the input value is a number, 2, we can use algebra to simplify.\r\n<div 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\\hfill \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>In this case, the input value is a letter so we cannot simplify the answer any further.\r\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div><\/li>\r\n \t<li>With an input value of [latex]a+h[\/latex], we must use the distributive property.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}f\\left(a+h\\right)&amp;={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4 \\\\[2mm] &amp;={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>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<div style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[\/latex]<\/div>\r\nand we know that\r\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\r\nNow we combine the results and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}&amp;=\\dfrac{\\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\\right)-\\left({a}^{2}+3a - 4\\right)}{h} \\\\[2mm] &amp;=\\dfrac{2ah+{h}^{2}+3h}{h}\\\\[2mm] &amp;=\\frac{h\\left(2a+h+3\\right)}{h}&amp;&amp;\\text{Factor out }h. \\\\[2mm] &amp;=2a+h+3&amp;&amp;\\text{Simplify}.\\end{align}[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1647&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], evaluate [latex]g\\left(5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"273881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"273881\"]\r\n\r\n[latex]g\\left(5\\right)=\\sqrt{5- 4}=1[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=97486&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/GLOmTED1UwA\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving Functions<\/h3>\r\nGiven the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], solve for [latex]h\\left(p\\right)=3[\/latex].\r\n\r\n[reveal-answer q=\"119909\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"119909\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;h\\left(p\\right)=3\\\\ &amp;{p}^{2}+2p=3 &amp;&amp;\\text{Substitute the original function }h\\left(p\\right)={p}^{2}+2p. \\\\ &amp;{p}^{2}+2p - 3=0 &amp;&amp;\\text{Subtract 3 from each side}. \\\\ &amp;\\left(p+3\\text{)(}p - 1\\right)=0 &amp;&amp;\\text{Factor}. \\end{align}[\/latex]<\/p>\r\nIf [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.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;p+3=0, &amp;&amp;p=-3 \\\\ &amp;p - 1=0, &amp;&amp;p=1\\hfill \\end{align}[\/latex]<\/p>\r\nThis 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].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190959\/CNX_Precalc_Figure_01_01_0062.jpg\" alt=\"Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).\" width=\"487\" height=\"459\" \/>\r\n\r\nWe can also verify by graphing as in Figure 5. The graph verifies that [latex]h\\left(1\\right)=h\\left(-3\\right)=3[\/latex] and [latex]h\\left(4\\right)=24[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=NTmgEF_nZSc\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], solve [latex]g\\left(m\\right)=2[\/latex].\r\n\r\n[reveal-answer q=\"480629\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"480629\"]\r\n\r\n[latex]m=8[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15766&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Evaluating Functions Expressed in Formulas<\/h2>\r\nSome functions are defined by mathematical rules or procedures expressed in <strong>equation<\/strong> form. If it is possible to express the function output with a <strong>formula<\/strong> 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]\u00a0and [latex]p[\/latex]. We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function in equation form, write its algebraic formula.<\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Finding an Equation of a Function<\/h3>\r\nExpress the relationship [latex]2n+6p=12[\/latex] as a function [latex]p=f\\left(n\\right)[\/latex], if possible.\r\n\r\n[reveal-answer q=\"938453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"938453\"]\r\n\r\nTo 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=[\/latex] expression involving [latex]n[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;2n+6p=12\\\\[1mm] &amp;6p=12 - 2n &amp;&amp;\\text{Subtract }2n\\text{ from both sides}. \\\\[1mm] &amp;p=\\frac{12 - 2n}{6} &amp;&amp;\\text{Divide both sides by 6 and simplify}. \\\\[1mm] &amp;p=\\frac{12}{6}-\\frac{2n}{6} \\\\[1mm] &amp;p=2-\\frac{1}{3}n \\end{align}[\/latex]<\/p>\r\nTherefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as\r\n<p style=\"text-align: center;\">[latex]p=f\\left(n\\right)=2-\\frac{1}{3}n[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nIt is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=lHTLjfPpFyQ\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Expressing the Equation of a Circle as a Function<\/h3>\r\nDoes 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].\r\n\r\n[reveal-answer q=\"557070\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"557070\"]\r\n\r\nFirst we subtract [latex]{x}^{2}[\/latex] from both sides.\r\n<p style=\"text-align: center;\">[latex]{y}^{2}=1-{x}^{2}[\/latex]<\/p>\r\nWe now try to solve for [latex]y[\/latex] in this equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\pm \\sqrt{1-{x}^{2}} \\\\[1mm] &amp;=\\sqrt{1-{x}^{2}}\\hspace{3mm}\\text{and}\\hspace{3mm}-\\sqrt{1-{x}^{2}} \\end{align}[\/latex]<\/p>\r\nWe 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIf [latex]x - 8{y}^{3}=0[\/latex], express [latex]y[\/latex] as a function of [latex]x[\/latex].\r\n\r\n[reveal-answer q=\"933974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"933974\"][latex]y=f\\left(x\\right)=\\cfrac{\\sqrt[3]{x}}{2}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111699&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?<\/strong>\r\n\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<h2>Evaluating a Function Given in Tabular Form<\/h2>\r\nAs we saw above, 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.\r\n\r\nThe 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 the table below.\r\n<table summary=\"Six rows and two columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th>Pet<\/th>\r\n<th>Memory span in hours<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Puppy<\/td>\r\n<td>0.008<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Adult dog<\/td>\r\n<td>0.083<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cat<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Goldfish<\/td>\r\n<td>2160<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Beta fish<\/td>\r\n<td>3600<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAt times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[\/latex].\r\n\r\nThe <strong>domain<\/strong> 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 \"goldfish.\" 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.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function represented by a table, identify specific output and input values.<strong>\r\n<\/strong><\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Evaluating and Solving a Tabular Function<\/h3>\r\nUsing the table below,\r\n<ol>\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 summary=\"Two rows and six columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"15206\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15206\"]\r\n<ul>\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. The table below shows two solutions: [latex]n=2[\/latex] and [latex]n=4[\/latex].<\/li>\r\n<\/ul>\r\n<table summary=\"Two rows and six columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the table from the previous example, evaluate [latex]g\\left(1\\right)[\/latex] .\r\n\r\n[reveal-answer q=\"724802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"724802\"][latex]g\\left(1\\right)=8[\/latex][\/hidden-answer]\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3751&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Finding Function Values from a Graph<\/h2>\r\nEvaluating 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).\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Reading Function Values from a Graph<\/h3>\r\nGiven the graph below,\r\n<ol>\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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191001\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/>\r\n[reveal-answer q=\"915833\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915833\"]\r\n<ol>\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 [latex]y[\/latex]-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex].<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/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]x=-1[\/latex] or [latex]x=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 the graph below.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191006\/CNX_Precalc_Figure_01_01_0092.jpg\" 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\" \/><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the graph, solve [latex]f\\left(x\\right)=1[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/>\r\n[reveal-answer q=\"529772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"529772\"]\r\n\r\n[latex]x=0[\/latex] or [latex]x=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom9\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2471&amp;theme=oea&amp;iframe_resize_id=mom9\" width=\"100%\" height=\"550\"><\/iframe>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2886&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"600\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Identify Functions Using Graphs<\/h2>\r\nAs we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis.\r\n\r\nThe most common 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 the graph below\u00a0tell 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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191012\/CNX_Precalc_Figure_01_01_0112.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" \/>\r\n\r\nThe <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. A vertical line includes all points with a particular [latex]x[\/latex] value. The [latex]y[\/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[\/latex] value. If we can draw <em>any<\/em> vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because that [latex]x[\/latex] value has more than one output. A function has only one output value for each input value.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191014\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a graph, use the vertical line test to determine if the graph represents a function.<\/h3>\r\n<ol>\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, the graph does not represent a function.<\/li>\r\n \t<li>If no vertical line can intersect the curve more than once, the graph does represent a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying the Vertical Line Test<\/h3>\r\nWhich of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/>\r\n[reveal-answer q=\"689864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"689864\"]\r\n\r\nIf 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 the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191020\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDoes the graph below represent a function?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/>\r\n[reveal-answer q=\"783855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783855\"]\r\n\r\nYes.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40676&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Identifying Basic Parent Functions<\/h2>\r\nIn this text, we explore functions\u2014the 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 \"parent functions,\" 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.\r\n\r\nWe will see these parent functions, combinations of parent functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these\u00a0parent 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 below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th colspan=\"3\">Parent Functions<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Name<\/th>\r\n<th>Function<\/th>\r\n<th>Graph<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Constant<\/td>\r\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191028\/CNX_Precalc_Figure_01_01_018n.jpg\" alt=\"Graph of a constant function.\" width=\"517\" height=\"319\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identity<\/td>\r\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191030\/CNX_Precalc_Figure_01_01_019n.jpg\" alt=\"Graph of a straight line.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Absolute value<\/td>\r\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191034\/CNX_Precalc_Figure_01_01_020n.jpg\" alt=\"Graph of absolute function.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quadratic<\/td>\r\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191037\/CNX_Precalc_Figure_01_01_021n.jpg\" alt=\"Graph of a parabola.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cubic<\/td>\r\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191039\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reciprocal\/ Rational<\/td>\r\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191042\/CNX_Precalc_Figure_01_01_023n.jpg\" alt=\"Graph of f(x)=1\/x.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reciprocal \/ Rational squared<\/td>\r\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191044\/CNX_Precalc_Figure_01_01_024n.jpg\" alt=\"Graph of f(x)=1\/x^2.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Square root<\/td>\r\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191047\/CNX_Precalc_Figure_01_01_025n.jpg\" alt=\"Graph of f(x)=sqrt(x).\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cube root<\/td>\r\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191050\/CNX_Precalc_Figure_01_01_026n.jpg\" alt=\"Graph of f(x)=x^(1\/3).\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111722&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\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 table 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<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137758543\" class=\"definition\">\r\n \t<dt><strong>dependent variable<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137758548\">an output variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137758552\" class=\"definition\">\r\n \t<dt><strong>domain<\/strong><\/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\" class=\"definition\">\r\n \t<dt><strong>function<\/strong><\/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\" class=\"definition\"><\/dl>\r\n<dl id=\"fs-id1165134149782\" class=\"definition\">\r\n \t<dt><strong>independent variable<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134149787\">an input variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135511353\" class=\"definition\">\r\n \t<dt><strong>input<\/strong><\/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\" class=\"definition\"><\/dl>\r\n<dl id=\"fs-id1165135508564\" class=\"definition\">\r\n \t<dt><strong>output<\/strong><\/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\" class=\"definition\">\r\n \t<dt><strong>range<\/strong><\/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\" class=\"definition\">\r\n \t<dt><strong>relation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135315542\" class=\"definition\">\r\n \t<dt><strong>vertical line test<\/strong><\/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<h2><\/h2>\r\n<img class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/>\r\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and Complete Homework 1.1 - Functions<\/span><\/h3>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Determine whether a relation represents a function.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Find function values.<\/span><\/li>\n<li>Use functional notation.<\/li>\n<li class=\"li2\"><span class=\"s1\">Use the vertical line test to identify functions.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Graph parent functions.<\/span><\/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<h2>Characteristics of Functions<\/h2>\n<p>A <strong>relation<\/strong> is a set of ordered pairs. The set of the first components of each <strong>ordered pair<\/strong> is called the <strong>domain <\/strong>of the relation\u00a0and the set of the second components of each ordered pair is called the <strong>range\u00a0<\/strong>of the relation. 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 the first.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(1,2\\right),\\left(2,4\\right),\\left(3,6\\right),\\left(4,8\\right),\\left(5,10\\right)\\right\\}[\/latex]<\/p>\n<p>The domain is [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0The range is [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].<\/p>\n<p>Note the values in the domain are also known as an <strong>input<\/strong> values, or values of the\u00a0<strong>independent variable<\/strong>, and are often labeled with the lowercase letter [latex]x[\/latex]. Values in the range are also known as an <strong>output<\/strong> values, or values of the\u00a0<strong>dependent variable<\/strong>, and are often labeled with the lowercase letter [latex]y[\/latex].<\/p>\n<p>A <strong>function<\/strong> [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 used more than once. For our example that relates the first five <strong>natural numbers<\/strong> to numbers double their values, this relation is a function because each element in the domain, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex], is paired with exactly one element in the range, [latex]\\left\\{2,4,6,8,10\\right\\}[\/latex].<\/p>\n<p>Now let\u2019s consider the set of ordered pairs that relates the terms &#8220;even&#8221; and &#8220;odd&#8221; to the first five natural numbers. It would appear as<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(\\text{odd},1\\right),\\left(\\text{even},2\\right),\\left(\\text{odd},3\\right),\\left(\\text{even},4\\right),\\left(\\text{odd},5\\right)\\right\\}[\/latex]<\/p>\n<p>Notice that each element in the domain, [latex]\\left\\{\\text{even,}\\text{odd}\\right\\}[\/latex]\u00a0is <em>not<\/em> paired with exactly one element in the range, [latex]\\left\\{1,2,3,4,5\\right\\}[\/latex].\u00a0For example, the term &#8220;odd&#8221; corresponds to three values from the domain, [latex]\\left\\{1,3,5\\right\\}[\/latex]\u00a0and the term &#8220;even&#8221; corresponds to two values from the range, [latex]\\left\\{2,4\\right\\}[\/latex].\u00a0This violates the definition of a function, so this relation is not a function.<\/p>\n<p>This image compares relations that are functions and not functions.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190946\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/><\/p>\n<p class=\"wp-caption-text\">(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 class=\"textbox\">\n<h3>A General Note: FunctionS<\/h3>\n<p>A <strong>function<\/strong> is a relation in which each possible input value leads to exactly one output value. We say &#8220;the output is a function of the input.&#8221;<\/p>\n<p>The <strong>input<\/strong> values make up the <strong>domain<\/strong>, and the <strong>output<\/strong> values make up the <strong>range<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3><strong>How To: Given a relationship between two quantities, determine whether the relationship is a function.<\/strong><\/h3>\n<ol>\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, the relationship is a function. If any input value leads to two or more outputs, the relationship is not a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining If Menu Price Lists Are Functions<\/h3>\n<p>The coffee shop menu consists of items and their prices.<\/p>\n<ol>\n<li>Is price a function of the item?<\/li>\n<li>Is the item a function of the price?<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190949\/CNX_Precalc_Figure_01_01_0042.jpg\" 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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q507796\">Show Solution<\/span><\/p>\n<div id=\"q507796\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Let\u2019s begin by considering the input as the items on the menu. The output values are then the prices.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190951\/CNX_Precalc_Figure_01_01_0272.jpg\" 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\" \/>Each item on the menu has only one price, so the price is a function of the item.<\/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.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190954\/CNX_Precalc_Figure_01_01_0282.jpg\" alt=\"Association of the prices to the donuts.\" width=\"731\" height=\"241\" \/>Therefore, the item is a not a function of price.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining If Class Grade Rules Are Functions<\/h3>\n<p>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? The table below shows a possible rule for assigning grade points.<\/p>\n<table>\n<tbody>\n<tr>\n<th>Percent Grade<\/th>\n<td>0\u201356<\/td>\n<td>57\u201361<\/td>\n<td>62\u201366<\/td>\n<td>67\u201371<\/td>\n<td>72\u201377<\/td>\n<td>78\u201386<\/td>\n<td>87\u201391<\/td>\n<td>92\u2013100<\/td>\n<\/tr>\n<tr>\n<th>Grade Point Average<\/th>\n<td>0.0<\/td>\n<td>1.0<\/td>\n<td>1.5<\/td>\n<td>2.0<\/td>\n<td>2.5<\/td>\n<td>3.0<\/td>\n<td>3.5<\/td>\n<td>4.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q813427\">Show Solution<\/span><\/p>\n<div id=\"q813427\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>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<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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The table below\u00a0lists the five greatest baseball players of all time in order of rank.<\/p>\n<table>\n<thead>\n<tr>\n<th>Player<\/th>\n<th>Rank<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Babe Ruth<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>Willie Mays<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>Ty Cobb<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>Walter Johnson<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>Hank Aaron<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q112010\">Show Solution<\/span><\/p>\n<div id=\"q112010\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>yes<\/li>\n<li>yes.\u00a0(Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=111625&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Using Function Notation<\/h3>\n<p>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 <strong>function notation<\/strong> is one representation that makes it easier to work\u00a0with functions.<\/p>\n<p>To represent &#8220;height is a function of age,&#8221; we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age. The letters [latex]f,g[\/latex], and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,y[\/latex], and [latex]z[\/latex] to represent numbers and [latex]A,B[\/latex],\u00a0and [latex]C[\/latex] to represent sets.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&h\\text{ is }f\\text{ of }a &&\\text{We name the function }f;\\text{ height is a function of age}. \\\\ &h=f\\left(a\\right) &&\\text{We use parentheses to indicate the function input}\\text{. } \\\\ &f\\left(a\\right) &&\\text{We name the function }f;\\text{ the expression is read as }\"f\\text{ of }a\". \\end{align}[\/latex]<\/p>\n<p>Remember, we can use any letter to name the function; we can use the notation [latex]h\\left(a\\right)[\/latex]\u00a0 to show that [latex]h[\/latex] depends on [latex]a[\/latex]. The input value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get an output value. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\n<p>We can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means &#8220;first add [latex]a[\/latex]\u00a0and [latex]b[\/latex], and the result is the input for the function [latex]f[\/latex].&#8221; We must perform the operations in this order to obtain the correct result.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Function Notation<\/h3>\n<p>The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as [latex]\"y[\/latex] is a function of [latex]x.\"[\/latex] The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex], or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Function Notation for Days in a Month<\/h3>\n<p>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 in a non-leap year.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q349740\">Show Solution<\/span><\/p>\n<div id=\"q349740\" class=\"hidden-answer\" style=\"display: none\">\n<p>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]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a &#8220;rule&#8221; that associates a specific number (the output) with each input.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190956\/CNX_Precalc_Figure_01_01_0052.jpg\" 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>For example, [latex]f\\left(\\text{April}\\right)=30[\/latex], because April has 30 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<h4>Analysis of the Solution<\/h4>\n<p>We must restrict the function to non-leap years. Otherwise February would have 2 outputs and this would not be a function. Also 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 as inputs and outputs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Interpreting Function Notation<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q299999\">Show Solution<\/span><\/p>\n<div id=\"q299999\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2510&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><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 &#8220;<em>y<\/em> is a function of <em>x<\/em>?&#8221;<\/strong><\/p>\n<p><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<h2>Representing Functions Using Tables<\/h2>\n<p>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.\u00a0In 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>The table below 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 summary=\"Two rows and thirteen columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Month number, [latex]m[\/latex] (input)<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td>9<\/td>\n<td>10<\/td>\n<td>11<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td><strong>Days in month, [latex]D[\/latex] (output)<\/strong><\/td>\n<td>31<\/td>\n<td>28<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The table below\u00a0defines 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[\/latex].<\/p>\n<table summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td>[latex]n[\/latex]<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>[latex]Q[\/latex]<\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The table below 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 summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Age in years, [latex]\\text{ }a\\text{ }[\/latex] (input)<\/strong><\/td>\n<td>5<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td>9<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td><strong>Height in inches, [latex]\\text{ }h\\text{ }[\/latex] (output)<\/strong><\/td>\n<td>40<\/td>\n<td>42<\/td>\n<td>44<\/td>\n<td>47<\/td>\n<td>50<\/td>\n<td>52<\/td>\n<td>54<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3><strong>How To: Given a table of input and output values, determine whether the table represents a function.<br \/>\n<\/strong><\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Identifying Tables that Represent Functions<\/h3>\n<p>Which table, A, B, or C, represents a function (if any)?<\/p>\n<table summary=\"Four rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Table A<\/th>\n<\/tr>\n<tr>\n<th>Input<\/th>\n<th>Output<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table summary=\"Four rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Table B<\/th>\n<\/tr>\n<tr>\n<th>Input<\/th>\n<th>Output<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u20133<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table summary=\"Four rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Table C<\/th>\n<\/tr>\n<tr>\n<th>Input<\/th>\n<th>Output<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q979211\">Show Solution<\/span><\/p>\n<div id=\"q979211\" class=\"hidden-answer\" style=\"display: none\">\n<p>a)\u00a0and b)\u00a0define functions. In both, each input value corresponds to exactly one output value. c)\u00a0does not define a function because the input value of 5 corresponds to two different output values.<\/p>\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)\u00a0can be represented by writing<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=1,f\\left(5\\right)=3,\\text{and }f\\left(8\\right)=6[\/latex]<\/p>\n<p>Similarly, the statements\u00a0[latex]g\\left(-3\\right)=5,g\\left(0\\right)=1,\\text{and }g\\left(4\\right)=5[\/latex]\u00a0represent the function in b).<\/p>\n<p>c)\u00a0cannot be expressed in a similar way because it does not represent a function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When we know an input value and want to determine the corresponding output value for a function, we <em>evaluate<\/em> the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.<\/p>\n<p>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 <em>solve<\/em> for the input. Solving can produce more than one solution because different input values can produce the same output value.<\/p>\n<h2>Evaluating and Solving Functions<\/h2>\n<p>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 class=\"textbox\">\n<h3><strong>How To: EVALUATE A FUNCTION Given ITS FORMula.<br \/>\n<\/strong><\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Evaluating Functions<\/h3>\n<p>Given the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], evaluate [latex]h\\left(4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q768180\">Show Solution<\/span><\/p>\n<div id=\"q768180\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">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 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>Therefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Evaluating Functions Using Function Notation (L9.3)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/Ehkzu5Uv7O0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Functions at Specific Values<\/h3>\n<p>For the function, [latex]f\\left(x\\right)={x}^{2}+3x - 4[\/latex], evaluate each of the following.<\/p>\n<ol>\n<li>[latex]f\\left(2\\right)[\/latex]<\/li>\n<li>[latex]f(a)[\/latex]<\/li>\n<li>[latex]f(a+h)[\/latex]<\/li>\n<li>[latex]\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q645951\">Show Solution<\/span><\/p>\n<div id=\"q645951\" class=\"hidden-answer\" style=\"display: none\">\n<p>Replace the [latex]x[\/latex]\u00a0in the function with each specified value.<\/p>\n<ol>\n<li>Because the input value is a number, 2, we can use algebra to simplify.\n<div style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&={2}^{2}+3\\left(2\\right)-4 \\\\ &=4+6 - 4 \\\\ &=6\\hfill \\end{align}[\/latex]<\/div>\n<\/li>\n<li>In this case, the input value is a letter so we cannot simplify the answer any further.\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\n<\/li>\n<li>With an input value of [latex]a+h[\/latex], we must use the distributive property.\n<div style=\"text-align: center;\">[latex]\\begin{align}f\\left(a+h\\right)&={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4 \\\\[2mm] &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>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<div style=\"text-align: center;\">[latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[\/latex]<\/div>\n<p>and we know that<\/p>\n<div style=\"text-align: center;\">[latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\n<p>Now we combine the results and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{f\\left(a+h\\right)-f\\left(a\\right)}{h}&=\\dfrac{\\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\\right)-\\left({a}^{2}+3a - 4\\right)}{h} \\\\[2mm] &=\\dfrac{2ah+{h}^{2}+3h}{h}\\\\[2mm] &=\\frac{h\\left(2a+h+3\\right)}{h}&&\\text{Factor out }h. \\\\[2mm] &=2a+h+3&&\\text{Simplify}.\\end{align}[\/latex]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1647&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], evaluate [latex]g\\left(5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q273881\">Show Solution<\/span><\/p>\n<div id=\"q273881\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g\\left(5\\right)=\\sqrt{5- 4}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=97486&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Find Function Inputs for a Given Quadratic Function Output\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GLOmTED1UwA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Functions<\/h3>\n<p>Given the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], solve for [latex]h\\left(p\\right)=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q119909\">Show Solution<\/span><\/p>\n<div id=\"q119909\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}&h\\left(p\\right)=3\\\\ &{p}^{2}+2p=3 &&\\text{Substitute the original function }h\\left(p\\right)={p}^{2}+2p. \\\\ &{p}^{2}+2p - 3=0 &&\\text{Subtract 3 from each side}. \\\\ &\\left(p+3\\text{)(}p - 1\\right)=0 &&\\text{Factor}. \\end{align}[\/latex]<\/p>\n<p>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<p style=\"text-align: center;\">[latex]\\begin{align}&p+3=0, &&p=-3 \\\\ &p - 1=0, &&p=1\\hfill \\end{align}[\/latex]<\/p>\n<p>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].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18190959\/CNX_Precalc_Figure_01_01_0062.jpg\" alt=\"Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).\" width=\"487\" height=\"459\" \/><\/p>\n<p>We can also verify by graphing as in Figure 5. 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>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Finding Function Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/NTmgEF_nZSc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]g\\left(m\\right)=\\sqrt{m - 4}[\/latex], solve [latex]g\\left(m\\right)=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q480629\">Show Solution<\/span><\/p>\n<div id=\"q480629\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]m=8[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15766&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluating Functions Expressed in Formulas<\/h2>\n<p>Some functions are defined by mathematical rules or procedures expressed in <strong>equation<\/strong> form. If it is possible to express the function output with a <strong>formula<\/strong> 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]\u00a0and [latex]p[\/latex]. We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a function in equation form, write its algebraic formula.<\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Finding an Equation of a Function<\/h3>\n<p>Express the relationship [latex]2n+6p=12[\/latex] as a function [latex]p=f\\left(n\\right)[\/latex], if possible.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q938453\">Show Solution<\/span><\/p>\n<div id=\"q938453\" class=\"hidden-answer\" style=\"display: none\">\n<p>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=[\/latex] expression involving [latex]n[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&2n+6p=12\\\\[1mm] &6p=12 - 2n &&\\text{Subtract }2n\\text{ from both sides}. \\\\[1mm] &p=\\frac{12 - 2n}{6} &&\\text{Divide both sides by 6 and simplify}. \\\\[1mm] &p=\\frac{12}{6}-\\frac{2n}{6} \\\\[1mm] &p=2-\\frac{1}{3}n \\end{align}[\/latex]<\/p>\n<p>Therefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as<\/p>\n<p style=\"text-align: center;\">[latex]p=f\\left(n\\right)=2-\\frac{1}{3}n[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex: Write a Linear Relation as a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lHTLjfPpFyQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Expressing the Equation of a Circle as a Function<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q557070\">Show Solution<\/span><\/p>\n<div id=\"q557070\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we subtract [latex]{x}^{2}[\/latex] from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]{y}^{2}=1-{x}^{2}[\/latex]<\/p>\n<p>We now try to solve for [latex]y[\/latex] in this equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\pm \\sqrt{1-{x}^{2}} \\\\[1mm] &=\\sqrt{1-{x}^{2}}\\hspace{3mm}\\text{and}\\hspace{3mm}-\\sqrt{1-{x}^{2}} \\end{align}[\/latex]<\/p>\n<p>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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>If [latex]x - 8{y}^{3}=0[\/latex], express [latex]y[\/latex] as a function of [latex]x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933974\">Show Solution<\/span><\/p>\n<div id=\"q933974\" class=\"hidden-answer\" style=\"display: none\">[latex]y=f\\left(x\\right)=\\cfrac{\\sqrt[3]{x}}{2}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111699&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><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<h2>Evaluating a Function Given in Tabular Form<\/h2>\n<p>As we saw above, 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>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 the table below.<\/p>\n<table summary=\"Six rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Pet<\/th>\n<th>Memory span in hours<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Puppy<\/td>\n<td>0.008<\/td>\n<\/tr>\n<tr>\n<td>Adult dog<\/td>\n<td>0.083<\/td>\n<\/tr>\n<tr>\n<td>Cat<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>Goldfish<\/td>\n<td>2160<\/td>\n<\/tr>\n<tr>\n<td>Beta fish<\/td>\n<td>3600<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[\/latex].<\/p>\n<p>The <strong>domain<\/strong> 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 &#8220;goldfish.&#8221; 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 class=\"textbox\">\n<h3>How To: Given a function represented by a table, identify specific output and input values.<strong><br \/>\n<\/strong><\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Evaluating and Solving a Tabular Function<\/h3>\n<p>Using the table below,<\/p>\n<ol>\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 summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15206\">Show Solution<\/span><\/p>\n<div id=\"q15206\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\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. The table below shows two solutions: [latex]n=2[\/latex] and [latex]n=4[\/latex].<\/li>\n<\/ul>\n<table summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]n[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g(n)[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the table from the previous example, evaluate [latex]g\\left(1\\right)[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q724802\">Show Solution<\/span><\/p>\n<div id=\"q724802\" class=\"hidden-answer\" style=\"display: none\">[latex]g\\left(1\\right)=8[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3751&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<h2>Finding Function Values from a Graph<\/h2>\n<p>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 class=\"textbox exercises\">\n<h3>Example: Reading Function Values from a Graph<\/h3>\n<p>Given the graph below,<\/p>\n<ol>\n<li>Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\n<li>Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191001\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q915833\">Show Solution<\/span><\/p>\n<div id=\"q915833\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the [latex]y[\/latex]-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/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]x=-1[\/latex] or [latex]x=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 the graph below.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191006\/CNX_Precalc_Figure_01_01_0092.jpg\" 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\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the graph, solve [latex]f\\left(x\\right)=1[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q529772\">Show Solution<\/span><\/p>\n<div id=\"q529772\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=0[\/latex] or [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom9\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2471&amp;theme=oea&amp;iframe_resize_id=mom9\" width=\"100%\" height=\"550\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2886&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"600\"><\/iframe><\/p>\n<\/div>\n<h2>Identify Functions Using Graphs<\/h2>\n<p>As we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\n<p>The most common 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 the graph below\u00a0tell 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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191012\/CNX_Precalc_Figure_01_01_0112.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" \/><\/p>\n<p>The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. A vertical line includes all points with a particular [latex]x[\/latex] value. The [latex]y[\/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[\/latex] value. If we can draw <em>any<\/em> vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because that [latex]x[\/latex] value has more than one output. A function has only one output value for each input value.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191014\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given a graph, use the vertical line test to determine if the graph represents a function.<\/h3>\n<ol>\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, the graph does not represent a function.<\/li>\n<li>If no vertical line can intersect the curve more than once, the graph does represent a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying the Vertical Line Test<\/h3>\n<p>Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q689864\">Show Solution<\/span><\/p>\n<div id=\"q689864\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191020\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Does the graph below represent a function?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783855\">Show Solution<\/span><\/p>\n<div id=\"q783855\" class=\"hidden-answer\" style=\"display: none\">\n<p>Yes.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40676&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<h2>Identifying Basic Parent Functions<\/h2>\n<p>In this text, we explore functions\u2014the 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 &#8220;parent functions,&#8221; 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>We will see these parent functions, combinations of parent functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these\u00a0parent 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 below.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"3\">Parent Functions<\/th>\n<\/tr>\n<tr>\n<th>Name<\/th>\n<th>Function<\/th>\n<th>Graph<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Constant<\/td>\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191028\/CNX_Precalc_Figure_01_01_018n.jpg\" alt=\"Graph of a constant function.\" width=\"517\" height=\"319\" \/><\/td>\n<\/tr>\n<tr>\n<td>Identity<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191030\/CNX_Precalc_Figure_01_01_019n.jpg\" alt=\"Graph of a straight line.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Absolute value<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191034\/CNX_Precalc_Figure_01_01_020n.jpg\" alt=\"Graph of absolute function.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Quadratic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191037\/CNX_Precalc_Figure_01_01_021n.jpg\" alt=\"Graph of a parabola.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Cubic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191039\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Reciprocal\/ Rational<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191042\/CNX_Precalc_Figure_01_01_023n.jpg\" alt=\"Graph of f(x)=1\/x.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Reciprocal \/ Rational squared<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191044\/CNX_Precalc_Figure_01_01_024n.jpg\" alt=\"Graph of f(x)=1\/x^2.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Square root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191047\/CNX_Precalc_Figure_01_01_025n.jpg\" alt=\"Graph of f(x)=sqrt(x).\" \/><\/td>\n<\/tr>\n<tr>\n<td>Cube root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191050\/CNX_Precalc_Figure_01_01_026n.jpg\" alt=\"Graph of f(x)=x^(1\/3).\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111722&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<h2>Key Concepts<\/h2>\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 table 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<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137758543\" class=\"definition\">\n<dt><strong>dependent variable<\/strong><\/dt>\n<dd id=\"fs-id1165137758548\">an output variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137758552\" class=\"definition\">\n<dt><strong>domain<\/strong><\/dt>\n<dd id=\"fs-id1165137932576\">the set of all possible input values for a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932580\" class=\"definition\">\n<dt><strong>function<\/strong><\/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\" class=\"definition\"><\/dl>\n<dl id=\"fs-id1165134149782\" class=\"definition\">\n<dt><strong>independent variable<\/strong><\/dt>\n<dd id=\"fs-id1165134149787\">an input variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511353\" class=\"definition\">\n<dt><strong>input<\/strong><\/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\" class=\"definition\"><\/dl>\n<dl id=\"fs-id1165135508564\" class=\"definition\">\n<dt><strong>output<\/strong><\/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\" class=\"definition\">\n<dt><strong>range<\/strong><\/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\" class=\"definition\">\n<dt><strong>relation<\/strong><\/dt>\n<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n<dt><strong>vertical line test<\/strong><\/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<h2><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/><\/p>\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and Complete Homework 1.1 &#8211; Functions<\/span><\/h3>\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-1780\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 111625, 111715, 11722. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 111699. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Function Notation Application. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g\">https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Function Notation Application. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g\">https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 2510, 1729. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15800. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1647. <strong>Authored by<\/strong>: WebWork-Rochester, mb Lippman,David, mb Sousa,James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 97486. <strong>Authored by<\/strong>: Carmichael, Patrick. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15766, 2886, 3751. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 40676. <strong>Authored by<\/strong>: Micheal Jenck. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Determine if a Relation is a Function. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zT69oxcMhPw\">https:\/\/youtu.be\/zT69oxcMhPw<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine if a Relation is a Function\",\"author\":\"James Sousa\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/zT69oxcMhPw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"cc\",\"description\":\"Function Notation Application\",\"author\":\"James Sousa\",\"organization\":\"\",\"url\":\"https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Function Notation Application\",\"author\":\"James Sousa\",\"organization\":\"\",\"url\":\"https:\/\/www.youtube.com\/watch?v=nAF_GZFwU1g\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 111625, 111715, 11722\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 2510, 1729\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 15800\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 1647\",\"author\":\"WebWork-Rochester, mb Lippman,David, mb Sousa,James\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 97486\",\"author\":\"Carmichael, Patrick\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 15766, 2886, 3751\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Question ID 111699\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 40676\",\"author\":\"Micheal Jenck\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1780","chapter","type-chapter","status-publish","hentry"],"part":1778,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1780","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/users\/708740"}],"version-history":[{"count":16,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1780\/revisions"}],"predecessor-version":[{"id":2574,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1780\/revisions\/2574"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/parts\/1778"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1780\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/media?parent=1780"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1780"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/contributor?post=1780"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/license?post=1780"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}