{"id":123,"date":"2023-06-21T13:22:35","date_gmt":"2023-06-21T13:22:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/characteristics-of-functions-and-their-graphs\/"},"modified":"2023-08-21T21:22:05","modified_gmt":"2023-08-21T21:22:05","slug":"characteristics-of-functions-and-their-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/characteristics-of-functions-and-their-graphs\/","title":{"raw":"\u25aa   Characteristics of Functions","rendered":"\u25aa   Characteristics of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify functions given in tabular form.<\/li>\r\n \t<li>Write and evaluate functional relationships using standard notation.<\/li>\r\n \t<li>Determine whether a function is one-to-one.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>about the notation<\/h3>\r\nRecall that ordered pairs of the form [latex]\\left(x, y\\right)[\/latex] contain elements from two different sets of values,\u00a0<em>x<\/em>-values and\u00a0<em>y<\/em>-values.\r\n\r\nWe use curly braces to denote a\u00a0<em>set<\/em> of elements. The set of elements named\u00a0<em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em> can be written as [latex]\\{a, b, c\\}[\/latex].\r\n\r\n<\/div>\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<\/strong>\u00a0<strong>of the relation<\/strong>. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice 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\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\r\n[ohm_question height=\"330\"]111625[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using Function Notation<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>recall function notation<\/h3>\r\nSome of the information in this section may look familiar from the Review for Success: Function Notation and Graphs of Functions. Learning to work with functions is naturally challenging, so it's worth another look at it. Don't be discouraged if the notation feels awkward or difficult to understand. Learning mathematics is very similar to learning a new language; it's normal that it takes lots of repetition before it starts to feel natural.\r\n\r\n<\/div>\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 with 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] 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.\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[ohm_question height=\"320\"]2510[\/ohm_question]\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 examples\">\r\n<h3>reading functions from a table<\/h3>\r\nNote that the tables above indicate which values are the inputs and which are the outputs to make it easier to clearly identify the relationship. Sometimes, though, the relationship is not labeled. For instance, if a table consists of measures of distance in one column and measures of time in another, you might fairly assume that time is the input and that a distance traveled would be dependent upon the time spent traveling. Either way, when reading a function from a table, it's a good idea to try to discern which values are intended to be or make the most sense as the inputs and which make most sense as outcome values, dependent upon the inputs.\r\n\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[ohm_question height=\"370\"]1729[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>recall evaluating functions<\/h3>\r\nSince a function [latex]y=f(x)[\/latex] is an equation, we can evaluate it for a numerical or algebraic input.\r\n\r\nEx. if [latex]f(x)=5x-11[\/latex], we can find [latex]f(2)[\/latex] or [latex]f(a+b)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f(2) &amp;= 5(2) -11 \\\\ &amp;= 10 - 11 \\\\ &amp;= -1\\end{align}[\/latex]<\/p>\r\n<p style=\"padding-left: 60px; text-align: center;\">or<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f(a+b) &amp;= 5(a+b)-11 \\\\ &amp;= 5a + 5b - 11\\end{align}[\/latex].<\/p>\r\nNote that we can also solve for what value of the input makes the output a certain value. That is, given our function above, [latex]f(x)=5x-11[\/latex], what value of\u00a0<em>x<\/em> makes [latex]f(x)=1[\/latex]?\r\n\r\n[reveal-answer q=\"252860\"]more[\/reveal-answer]\r\n[hidden-answer a=\"252860\"] The idea is to replace [latex]f(x)[\/latex] the value of 1, then solve for the input\u00a0<em>x<\/em>\u00a0that makes that a true statement.\r\n\r\n[latex]\\begin{align}f(x) &amp;= 5x-11 \\\\ 1 &amp;= 5x-11 \\\\ 12 &amp;= 5x \\\\ \\dfrac{12}{5} &amp;= x\\end{align}[\/latex]\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<h2>Determine whether a function is one-to-one<\/h2>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191009\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" \/>\r\n\r\nSome functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.\r\n\r\nHowever, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in.\r\n<table summary=\"Two columns and five rows. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th>Letter grade<\/th>\r\n<th>Grade point average<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>A<\/td>\r\n<td>4.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B<\/td>\r\n<td>3.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C<\/td>\r\n<td>2.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<\/td>\r\n<td>1.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.\r\n\r\nTo visualize this concept, let\u2019s look again at the two simple functions sketched in (a) and (b)\u00a0below.\r\n\r\n<img class=\"aligncenter\" 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\" \/>\r\n\r\nThe function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.\r\n<div class=\"textbox\">\r\n<h3>A General Note: One-to-One Function<\/h3>\r\nA one-to-one function is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining Whether a Relationship Is a One-to-One Function<\/h3>\r\nIs the area of a circle a function of its radius? If yes, is the function one-to-one?\r\n\r\n[reveal-answer q=\"380432\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380432\"]\r\n\r\nA circle of radius [latex]r[\/latex] has a unique area measure given by [latex]A=\\pi {r}^{2}[\/latex], so for any input, [latex]r[\/latex], there is only one output, [latex]A[\/latex]. The area is a function of radius [latex]r[\/latex].\r\n\r\nIf the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[\/latex] is given by the formula [latex]A=\\pi {r}^{2}[\/latex]. Because areas and radii are positive numbers, there is exactly one solution: [latex]r=\\sqrt{\\frac{A}{\\pi }}[\/latex]. So the area of a circle is a one-to-one function of the circle\u2019s radius.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>recall manipulating formulas to isolate a certain variable<\/h3>\r\nIn the example above, a formula was solved for one of the variables. We often need to do this in order to rewrite the formula in a useful way.\r\n\r\n[reveal-answer q=\"639172\"]more[\/reveal-answer]\r\n[hidden-answer a=\"639172\"]\r\n\r\nIn the example above, the formula for the area of a circle,\u00a0[latex]A=\\pi {r}^{2}[\/latex], was solved for A using the properties of equality.\r\n\r\n[latex]\\begin{align}A &amp;= \\pi {r}^{2} \\\\ \\dfrac{A}{\\pi} &amp;= r^2 \\quad \\text{divide both sides by }\\pi \\\\ r &amp;= \\sqrt{\\frac{A}{\\pi }} \\quad \\text{ take the square root of both sides}\\end{align}[\/latex]\r\n\r\nOnly the positive root was required since radius cannot be negative.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<ol>\r\n \t<li>Is a balance a function of the bank account number?<\/li>\r\n \t<li>Is a bank account number a function of the balance?<\/li>\r\n \t<li>Is a balance a one-to-one function of the bank account number?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"997233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"997233\"]\r\n<ol>\r\n \t<li><span class=\"s1\">yes, because each bank account (input) has a single balance (output) at any given time.<\/span><\/li>\r\n \t<li><span class=\"s1\">no, because several bank accounts (inputs) may have the same balance (output).<\/span><\/li>\r\n \t<li><span class=\"s1\">no, because the more than one bank account (input) can have the same balance (output).<\/span><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"380\"]15800[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify functions given in tabular form.<\/li>\n<li>Write and evaluate functional relationships using standard notation.<\/li>\n<li>Determine whether a function is one-to-one.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>about the notation<\/h3>\n<p>Recall that ordered pairs of the form [latex]\\left(x, y\\right)[\/latex] contain elements from two different sets of values,\u00a0<em>x<\/em>-values and\u00a0<em>y<\/em>-values.<\/p>\n<p>We use curly braces to denote a\u00a0<em>set<\/em> of elements. The set of elements named\u00a0<em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em> can be written as [latex]\\{a, b, c\\}[\/latex].<\/p>\n<\/div>\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<\/strong>\u00a0<strong>of the relation<\/strong>. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice 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<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=\"ohm111625\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111625&theme=oea&iframe_resize_id=ohm111625&show_question_numbers\" width=\"100%\" height=\"330\"><\/iframe><\/p>\n<\/div>\n<h2>Using Function Notation<\/h2>\n<div class=\"textbox examples\">\n<h3>recall function notation<\/h3>\n<p>Some of the information in this section may look familiar from the Review for Success: Function Notation and Graphs of Functions. Learning to work with functions is naturally challenging, so it&#8217;s worth another look at it. Don&#8217;t be discouraged if the notation feels awkward or difficult to understand. Learning mathematics is very similar to learning a new language; it&#8217;s normal that it takes lots of repetition before it starts to feel natural.<\/p>\n<\/div>\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 with 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] 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.<\/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=\"ohm2510\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2510&theme=oea&iframe_resize_id=ohm2510&show_question_numbers\" width=\"100%\" height=\"320\"><\/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 examples\">\n<h3>reading functions from a table<\/h3>\n<p>Note that the tables above indicate which values are the inputs and which are the outputs to make it easier to clearly identify the relationship. Sometimes, though, the relationship is not labeled. For instance, if a table consists of measures of distance in one column and measures of time in another, you might fairly assume that time is the input and that a distance traveled would be dependent upon the time spent traveling. Either way, when reading a function from a table, it&#8217;s a good idea to try to discern which values are intended to be or make the most sense as the inputs and which make most sense as outcome values, dependent upon the inputs.<\/p>\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<p><iframe loading=\"lazy\" id=\"ohm1729\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1729&theme=oea&iframe_resize_id=ohm1729&show_question_numbers\" width=\"100%\" height=\"370\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>recall evaluating functions<\/h3>\n<p>Since a function [latex]y=f(x)[\/latex] is an equation, we can evaluate it for a numerical or algebraic input.<\/p>\n<p>Ex. if [latex]f(x)=5x-11[\/latex], we can find [latex]f(2)[\/latex] or [latex]f(a+b)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f(2) &= 5(2) -11 \\\\ &= 10 - 11 \\\\ &= -1\\end{align}[\/latex]<\/p>\n<p style=\"padding-left: 60px; text-align: center;\">or<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f(a+b) &= 5(a+b)-11 \\\\ &= 5a + 5b - 11\\end{align}[\/latex].<\/p>\n<p>Note that we can also solve for what value of the input makes the output a certain value. That is, given our function above, [latex]f(x)=5x-11[\/latex], what value of\u00a0<em>x<\/em> makes [latex]f(x)=1[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q252860\">more<\/span><\/p>\n<div id=\"q252860\" class=\"hidden-answer\" style=\"display: none\"> The idea is to replace [latex]f(x)[\/latex] the value of 1, then solve for the input\u00a0<em>x<\/em>\u00a0that makes that a true statement.<\/p>\n<p>[latex]\\begin{align}f(x) &= 5x-11 \\\\ 1 &= 5x-11 \\\\ 12 &= 5x \\\\ \\dfrac{12}{5} &= x\\end{align}[\/latex]<\/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>Determine whether a function is one-to-one<\/h2>\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\/18191009\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" \/><\/p>\n<p>Some functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.<\/p>\n<p>However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in.<\/p>\n<table summary=\"Two columns and five rows. The first column is labeled,\">\n<thead>\n<tr>\n<th>Letter grade<\/th>\n<th>Grade point average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>A<\/td>\n<td>4.0<\/td>\n<\/tr>\n<tr>\n<td>B<\/td>\n<td>3.0<\/td>\n<\/tr>\n<tr>\n<td>C<\/td>\n<td>2.0<\/td>\n<\/tr>\n<tr>\n<td>D<\/td>\n<td>1.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n<p>To visualize this concept, let\u2019s look again at the two simple functions sketched in (a) and (b)\u00a0below.<\/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\/18190946\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/><\/p>\n<p>The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: One-to-One Function<\/h3>\n<p>A one-to-one function is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining Whether a Relationship Is a One-to-One Function<\/h3>\n<p>Is the area of a circle a function of its radius? If yes, is the function one-to-one?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380432\">Show Solution<\/span><\/p>\n<div id=\"q380432\" class=\"hidden-answer\" style=\"display: none\">\n<p>A circle of radius [latex]r[\/latex] has a unique area measure given by [latex]A=\\pi {r}^{2}[\/latex], so for any input, [latex]r[\/latex], there is only one output, [latex]A[\/latex]. The area is a function of radius [latex]r[\/latex].<\/p>\n<p>If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[\/latex] is given by the formula [latex]A=\\pi {r}^{2}[\/latex]. Because areas and radii are positive numbers, there is exactly one solution: [latex]r=\\sqrt{\\frac{A}{\\pi }}[\/latex]. So the area of a circle is a one-to-one function of the circle\u2019s radius.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>recall manipulating formulas to isolate a certain variable<\/h3>\n<p>In the example above, a formula was solved for one of the variables. We often need to do this in order to rewrite the formula in a useful way.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q639172\">more<\/span><\/p>\n<div id=\"q639172\" class=\"hidden-answer\" style=\"display: none\">\n<p>In the example above, the formula for the area of a circle,\u00a0[latex]A=\\pi {r}^{2}[\/latex], was solved for A using the properties of equality.<\/p>\n<p>[latex]\\begin{align}A &= \\pi {r}^{2} \\\\ \\dfrac{A}{\\pi} &= r^2 \\quad \\text{divide both sides by }\\pi \\\\ r &= \\sqrt{\\frac{A}{\\pi }} \\quad \\text{ take the square root of both sides}\\end{align}[\/latex]<\/p>\n<p>Only the positive root was required since radius cannot be negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<ol>\n<li>Is a balance a function of the bank account number?<\/li>\n<li>Is a bank account number a function of the balance?<\/li>\n<li>Is a balance a one-to-one function of the bank account number?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q997233\">Show Solution<\/span><\/p>\n<div id=\"q997233\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><span class=\"s1\">yes, because each bank account (input) has a single balance (output) at any given time.<\/span><\/li>\n<li><span class=\"s1\">no, because several bank accounts (inputs) may have the same balance (output).<\/span><\/li>\n<li><span class=\"s1\">no, because the more than one bank account (input) can have the same balance (output).<\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm15800\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15800&theme=oea&iframe_resize_id=ohm15800&show_question_numbers\" width=\"100%\" height=\"380\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-123\">\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. <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>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>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>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><\/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":395986,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Determine if a Relation is a Function\",\"author\":\"James 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