{"id":625,"date":"2022-01-19T18:04:44","date_gmt":"2022-01-19T18:04:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=625"},"modified":"2025-12-08T18:33:12","modified_gmt":"2025-12-08T18:33:12","slug":"1-1-2-vertical-line-test-and-horizontal-line-test","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/1-1-2-vertical-line-test-and-horizontal-line-test\/","title":{"raw":"1.1.3: Vertical and Horizontal Line Tests","rendered":"1.1.3: Vertical and Horizontal Line Tests"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the vertical line test to determine if a graph represents a function<\/li>\r\n \t<li>Use the horizontal line test to determine if a graphed function is one-to-one<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Functions as Graphs<\/h2>\r\nAs we saw in the previous section, functions can be represented by graphs. Graphs are useful as they display many input-output pairs in a small space and the visual information they provide can make relationships clearer. We typically construct graphs with the domain values along the horizontal axis and the range values along the vertical axis. This is because the domain values (e.g. [latex]x[\/latex]) are\u00a0<em><strong>independent\u00a0<\/strong><\/em>and independent variables are generally associated with the horizontal axis. The values in the range are called <em><strong>function values<\/strong><\/em> and\u00a0are <em><strong>dependent<\/strong><\/em> on the domain values and are therefore associated with the vertical axis.\r\n\r\nSince the rectangular coordinate system is most commonly associated with an [latex]x[\/latex]-axis and a [latex]y[\/latex]-axis, along with coordinate pairs [latex](x, y)[\/latex], it is common when first learning functions to name the domain values [latex]x[\/latex] and the function values [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 [latex]x[\/latex]-coordinate of each point is a domain value and the [latex]y[\/latex]-coordinate of each point is the corresponding function value.\r\n\r\n<span style=\"font-size: 1em;\">In other words, if [latex]y=f(x)[\/latex], then the coordinates of any point on the graph of [latex]y=f(x)[\/latex] are of the form [latex]\\left (x, f(x)\\right )[\/latex].\u00a0<\/span>For example, the coordinates of the black dots on the graph in figure 1 are (0, 2) and (6, 1). This tells us that the function value when [latex]x=0[\/latex] is 2, and the function value when [latex]x=6[\/latex] is 1. In function notation, this is written [latex]f(0)=2[\/latex] and [latex]f(6)=1[\/latex]. The points [latex](0, 1)[\/latex] and [latex](6, 1)[\/latex] are only two points on the graph. The graph of the set of all points [latex]\\left(x,f(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[caption id=\"\" align=\"aligncenter\" width=\"368\"]<img class=\"\" 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=\"368\" height=\"222\" \/> Figure 1. Graph of a function[\/caption]\r\n<h2>Some Graphs Are Not Functions<\/h2>\r\nAny function [latex]y=f(x)[\/latex] can be represented by a graph. However, every graph does not represent a function. Remember that a function is a relation with a one-to-one or many-to-one mapping. All graphs represent relations, but that relation may not be a function.\r\n\r\nConsider the graph in figure 1. Choose any [latex]x[\/latex]-value and it is clear that it has a single corresponding [latex]y[\/latex]-value. There is no [latex]x[\/latex]-value that maps to two or more [latex]y[\/latex]-values. That is the definition of a function. Now consider the graph in figure 2.\r\n\r\n[caption id=\"attachment_727\" align=\"aligncenter\" width=\"490\"]<img class=\"wp-image-727\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11211045\/xy%5E2-4-1024x462.png\" alt=\"Graph the double backs on itself\" width=\"490\" height=\"221\" \/> Figure 2. Graph does not represent a function[\/caption]\r\n\r\nThe only [latex]x[\/latex]-value that has a single [latex]y[\/latex]-value is [latex]x=-4[\/latex]. Any other\u00a0[latex]x[\/latex]-value we choose has 2\u00a0[latex]y[\/latex]-values. This means that the mapping of\u00a0[latex]x[\/latex] to\u00a0[latex]y[\/latex] is one-to-many and therefore this graph does not represent a function.\r\n<h3>The Vertical Line Test<\/h3>\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. 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 [latex]y[\/latex]-value. For each [latex]x[\/latex]-value in the domain, a function has only one [latex]y[\/latex]-value (or function value) in the range.\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=\"860\" height=\"239\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>the vertical line test<\/h3>\r\nThe vertical line test can be used to determine of a graph represents a function.\r\n<ol>\r\n \t<li>Inspect the graph to see if <em>any<\/em> vertical line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If <em>any<\/em> vertical line intersects the graph more than once, the graph does NOT represent a function.<\/li>\r\n \t<li>If <em>any<\/em> vertical line intersects the graph only once, the graph does represent a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"examples\">\r\n<h3>Example 1<\/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=\"639\" height=\"274\" \/>\r\n<h4>Solution<\/h4>\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) above. From this we can conclude that these two graphs represent functions. The third graph, (c),\u00a0 does <em>not<\/em> represent a function because a vertical line intersects 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=\"321\" height=\"293\" \/>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Transcript-1.1.3-1.odt\">Transcript-1.1.3-1<\/a>\r\n<div class=\"try it\">\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDoes the graph 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=\"315\" height=\"237\" \/>\r\n\r\n[reveal-answer q=\"337792\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"337792\"]Yes[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"try it\">\r\n<div class=\"textbox tryit\">\r\n<h3>try it 2<\/h3>\r\nIn the graph below, use the vertical line test to determine which relations are functions and which are not.\r\n\r\n<img class=\"aligncenter wp-image-6721\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08191348\/Screen-Shot-2019-07-08-at-12.11.16-PM.png\" alt=\"Graph with a red circle, a blue line, an orange polynomial, and a green sideways parabola.\" width=\"279\" height=\"264\" \/>\r\n\r\n[reveal-answer q=\"709223\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"709223\"]\r\n\r\nRed circle: No\r\n\r\nOrange repeating curve: Yes\r\n\r\nBlue line: Yes\r\n\r\nGreen parabola: No\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>The Horizontal Line Test<\/h3>\r\nIn the previous section, we learned that a function is a relation with a one-to-one or many-to-one mapping. The vertical line test can tell us if a graph represents a function, but how will we know if that function is one-to-one?\u00a0 In order for a function to be one-to-one, not only does every\u00a0[latex]x[\/latex]-value map to only one\u00a0[latex]y[\/latex]-value but every\u00a0[latex]y[\/latex]-value is mapped from exactly one\u00a0[latex]x[\/latex]-value.\r\n\r\nConsider the graph in figure 3. We can see it is a function as any vertical line passes through only one point. But is the function one-to-one? If we choose any\u00a0[latex]y[\/latex]-value, say\u00a0[latex]y=4[\/latex], we can see that the points [latex](-2, 4)[\/latex] and [latex](2, 4)[\/latex] both lie on the curve. This means that this function is\u00a0<em>not<\/em> one-to-one as the\u00a0[latex]y[\/latex]-value 4 is mapped from two different\u00a0[latex]x[\/latex]-values, -2 and 2.\r\n\r\n[caption id=\"attachment_729\" align=\"aligncenter\" width=\"230\"]<img class=\"wp-image-729 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11213915\/yx%5E2-230x300.png\" alt=\"Graph showing a parabola\" width=\"230\" height=\"300\" \/> Figure 3. A function that is not one-to-one[\/caption]\r\n\r\nOnce we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the <em><strong>horizontal line test<\/strong><\/em>. A horizontal line includes all points with a particular [latex]y[\/latex]-value. If we can draw <em>any<\/em> horizontal line that intersects the graph more than once, then the graph does <em>not<\/em> represent a one-to-one function because that [latex]y[\/latex] value is mapped from more than one\u00a0[latex]x[\/latex]-value.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/h3>\r\n<ol>\r\n \t<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If there is any such line, the function is not one-to-one.<\/li>\r\n \t<li>If no horizontal line can intersect the curve more than once, the function is one-to-one.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"examples\">\r\n<h3>Example 2<\/h3>\r\nDetermine if the graph represents a one-to-one 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\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"A) graph of a polynomial with two turning points, B) graph of a line, and C) graph of a circle.\" width=\"683\" height=\"293\" \/>\r\n<h4>Solution<\/h4>\r\n<img class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191025\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"Graph of a polynomial with a horizontal line showing two intersections.\" width=\"236\" height=\"216\" \/>\r\n\r\n<span style=\"font-size: 1em;\">(a) The graph represents a function since any vertical line passes through only one point on the graph. But the function is not one-to-one. The horizontal line shown below intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/span>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n(b) The graph represents a one-to-one function as any vertical and horizontal lines pass through only one point on the graph.\r\n\r\n(c) The graph does not represent a function as a vertical line can pass through more than one point. Since it is not a function, it cannot be a one-to-one function.;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nDetermine if the graph represents a one-to-one function.\r\n\r\n[caption id=\"attachment_733\" align=\"alignleft\" width=\"233\"]<img class=\"wp-image-733 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11220155\/ysinx3-300x170.png\" alt=\"Graph of horizontal wave\" width=\"233\" height=\"132\" \/> Graph (a)[\/caption]\r\n\r\n[caption id=\"attachment_731\" align=\"alignleft\" width=\"206\"]<img class=\"wp-image-731 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11220146\/x%5E3-10-275x300.png\" alt=\"Graph of y=x^3\" width=\"206\" height=\"225\" \/> Graph (b)[\/caption]\r\n\r\n[caption id=\"attachment_732\" align=\"alignleft\" width=\"216\"]<img class=\"wp-image-732\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11220151\/xcosx-289x300.png\" alt=\"Graph of horizontal wave increasing in amplitude.\" width=\"216\" height=\"224\" \/> Graph (c)[\/caption]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"hjm403\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm403\"]\r\n\r\nGraph (a) is a function but is not one-to-one.\r\n\r\nGraph (b) is a one-to-one function.\r\n\r\nGraph (c) is a function but is not one-to-one.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the vertical line test to determine if a graph represents a function<\/li>\n<li>Use the horizontal line test to determine if a graphed function is one-to-one<\/li>\n<\/ul>\n<\/div>\n<h2>Functions as Graphs<\/h2>\n<p>As we saw in the previous section, functions can be represented by graphs. Graphs are useful as they display many input-output pairs in a small space and the visual information they provide can make relationships clearer. We typically construct graphs with the domain values along the horizontal axis and the range values along the vertical axis. This is because the domain values (e.g. [latex]x[\/latex]) are\u00a0<em><strong>independent\u00a0<\/strong><\/em>and independent variables are generally associated with the horizontal axis. The values in the range are called <em><strong>function values<\/strong><\/em> and\u00a0are <em><strong>dependent<\/strong><\/em> on the domain values and are therefore associated with the vertical axis.<\/p>\n<p>Since the rectangular coordinate system is most commonly associated with an [latex]x[\/latex]-axis and a [latex]y[\/latex]-axis, along with coordinate pairs [latex](x, y)[\/latex], it is common when first learning functions to name the domain values [latex]x[\/latex] and the function values [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 [latex]x[\/latex]-coordinate of each point is a domain value and the [latex]y[\/latex]-coordinate of each point is the corresponding function value.<\/p>\n<p><span style=\"font-size: 1em;\">In other words, if [latex]y=f(x)[\/latex], then the coordinates of any point on the graph of [latex]y=f(x)[\/latex] are of the form [latex]\\left (x, f(x)\\right )[\/latex].\u00a0<\/span>For example, the coordinates of the black dots on the graph in figure 1 are (0, 2) and (6, 1). This tells us that the function value when [latex]x=0[\/latex] is 2, and the function value when [latex]x=6[\/latex] is 1. In function notation, this is written [latex]f(0)=2[\/latex] and [latex]f(6)=1[\/latex]. The points [latex](0, 1)[\/latex] and [latex](6, 1)[\/latex] are only two points on the graph. The graph of the set of all points [latex]\\left(x,f(x)\\right)[\/latex] is a curve. The curve shown includes [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex] because the curve passes through those points.<\/p>\n<div style=\"width: 378px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" 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=\"368\" height=\"222\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Graph of a function<\/p>\n<\/div>\n<h2>Some Graphs Are Not Functions<\/h2>\n<p>Any function [latex]y=f(x)[\/latex] can be represented by a graph. However, every graph does not represent a function. Remember that a function is a relation with a one-to-one or many-to-one mapping. All graphs represent relations, but that relation may not be a function.<\/p>\n<p>Consider the graph in figure 1. Choose any [latex]x[\/latex]-value and it is clear that it has a single corresponding [latex]y[\/latex]-value. There is no [latex]x[\/latex]-value that maps to two or more [latex]y[\/latex]-values. That is the definition of a function. Now consider the graph in figure 2.<\/p>\n<div id=\"attachment_727\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-727\" class=\"wp-image-727\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11211045\/xy%5E2-4-1024x462.png\" alt=\"Graph the double backs on itself\" width=\"490\" height=\"221\" \/><\/p>\n<p id=\"caption-attachment-727\" class=\"wp-caption-text\">Figure 2. Graph does not represent a function<\/p>\n<\/div>\n<p>The only [latex]x[\/latex]-value that has a single [latex]y[\/latex]-value is [latex]x=-4[\/latex]. Any other\u00a0[latex]x[\/latex]-value we choose has 2\u00a0[latex]y[\/latex]-values. This means that the mapping of\u00a0[latex]x[\/latex] to\u00a0[latex]y[\/latex] is one-to-many and therefore this graph does not represent a function.<\/p>\n<h3>The Vertical Line Test<\/h3>\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. 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 [latex]y[\/latex]-value. For each [latex]x[\/latex]-value in the domain, a function has only one [latex]y[\/latex]-value (or function value) in the range.<\/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=\"860\" height=\"239\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>the vertical line test<\/h3>\n<p>The vertical line test can be used to determine of a graph represents a function.<\/p>\n<ol>\n<li>Inspect the graph to see if <em>any<\/em> vertical line drawn would intersect the curve more than once.<\/li>\n<li>If <em>any<\/em> vertical line intersects the graph more than once, the graph does NOT represent a function.<\/li>\n<li>If <em>any<\/em> vertical line intersects the graph only once, the graph does represent a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"examples\">\n<h3>Example 1<\/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=\"639\" height=\"274\" \/><\/p>\n<h4>Solution<\/h4>\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) above. From this we can conclude that these two graphs represent functions. The third graph, (c),\u00a0 does <em>not<\/em> represent a function because a vertical line intersects 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=\"321\" height=\"293\" \/><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Transcript-1.1.3-1.odt\">Transcript-1.1.3-1<\/a><\/p>\n<div class=\"try it\">\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Does the graph 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=\"315\" height=\"237\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q337792\">Show Answer<\/span><\/p>\n<div id=\"q337792\" class=\"hidden-answer\" style=\"display: none\">Yes<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"try it\">\n<div class=\"textbox tryit\">\n<h3>try it 2<\/h3>\n<p>In the graph below, use the vertical line test to determine which relations are functions and which are not.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6721\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08191348\/Screen-Shot-2019-07-08-at-12.11.16-PM.png\" alt=\"Graph with a red circle, a blue line, an orange polynomial, and a green sideways parabola.\" width=\"279\" height=\"264\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q709223\">Show Answer<\/span><\/p>\n<div id=\"q709223\" class=\"hidden-answer\" style=\"display: none\">\n<p>Red circle: No<\/p>\n<p>Orange repeating curve: Yes<\/p>\n<p>Blue line: Yes<\/p>\n<p>Green parabola: No<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>The Horizontal Line Test<\/h3>\n<p>In the previous section, we learned that a function is a relation with a one-to-one or many-to-one mapping. The vertical line test can tell us if a graph represents a function, but how will we know if that function is one-to-one?\u00a0 In order for a function to be one-to-one, not only does every\u00a0[latex]x[\/latex]-value map to only one\u00a0[latex]y[\/latex]-value but every\u00a0[latex]y[\/latex]-value is mapped from exactly one\u00a0[latex]x[\/latex]-value.<\/p>\n<p>Consider the graph in figure 3. We can see it is a function as any vertical line passes through only one point. But is the function one-to-one? If we choose any\u00a0[latex]y[\/latex]-value, say\u00a0[latex]y=4[\/latex], we can see that the points [latex](-2, 4)[\/latex] and [latex](2, 4)[\/latex] both lie on the curve. This means that this function is\u00a0<em>not<\/em> one-to-one as the\u00a0[latex]y[\/latex]-value 4 is mapped from two different\u00a0[latex]x[\/latex]-values, -2 and 2.<\/p>\n<div id=\"attachment_729\" style=\"width: 240px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-729\" class=\"wp-image-729 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11213915\/yx%5E2-230x300.png\" alt=\"Graph showing a parabola\" width=\"230\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-729\" class=\"wp-caption-text\">Figure 3. A function that is not one-to-one<\/p>\n<\/div>\n<p>Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the <em><strong>horizontal line test<\/strong><\/em>. A horizontal line includes all points with a particular [latex]y[\/latex]-value. If we can draw <em>any<\/em> horizontal line that intersects the graph more than once, then the graph does <em>not<\/em> represent a one-to-one function because that [latex]y[\/latex] value is mapped from more than one\u00a0[latex]x[\/latex]-value.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/h3>\n<ol>\n<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, the function is not one-to-one.<\/li>\n<li>If no horizontal line can intersect the curve more than once, the function is one-to-one.<\/li>\n<\/ol>\n<\/div>\n<div class=\"examples\">\n<h3>Example 2<\/h3>\n<p>Determine if the graph represents a one-to-one 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\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"A) graph of a polynomial with two turning points, B) graph of a line, and C) graph of a circle.\" width=\"683\" height=\"293\" \/><\/p>\n<h4>Solution<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191025\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"Graph of a polynomial with a horizontal line showing two intersections.\" width=\"236\" height=\"216\" \/><\/p>\n<p><span style=\"font-size: 1em;\">(a) The graph represents a function since any vertical line passes through only one point on the graph. But the function is not one-to-one. The horizontal line shown below intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>(b) The graph represents a one-to-one function as any vertical and horizontal lines pass through only one point on the graph.<\/p>\n<p>(c) The graph does not represent a function as a vertical line can pass through more than one point. Since it is not a function, it cannot be a one-to-one function.;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Determine if the graph represents a one-to-one function.<\/p>\n<div id=\"attachment_733\" style=\"width: 243px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-733\" class=\"wp-image-733\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11220155\/ysinx3-300x170.png\" alt=\"Graph of horizontal wave\" width=\"233\" height=\"132\" \/><\/p>\n<p id=\"caption-attachment-733\" class=\"wp-caption-text\">Graph (a)<\/p>\n<\/div>\n<div id=\"attachment_731\" style=\"width: 216px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-731\" class=\"wp-image-731\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11220146\/x%5E3-10-275x300.png\" alt=\"Graph of y=x^3\" width=\"206\" height=\"225\" \/><\/p>\n<p id=\"caption-attachment-731\" class=\"wp-caption-text\">Graph (b)<\/p>\n<\/div>\n<div id=\"attachment_732\" style=\"width: 226px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-732\" class=\"wp-image-732\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/11220151\/xcosx-289x300.png\" alt=\"Graph of horizontal wave increasing in amplitude.\" width=\"216\" height=\"224\" \/><\/p>\n<p id=\"caption-attachment-732\" class=\"wp-caption-text\">Graph (c)<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm403\">Show Answer<\/span><\/p>\n<div id=\"qhjm403\" class=\"hidden-answer\" style=\"display: none\">\n<p>Graph (a) is a function but is not one-to-one.<\/p>\n<p>Graph (b) is a one-to-one function.<\/p>\n<p>Graph (c) is a function but is not one-to-one.<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; 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