{"id":15965,"date":"2019-12-05T05:17:04","date_gmt":"2019-12-05T05:17:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/using-the-vertical-line-test\/"},"modified":"2019-12-05T05:17:35","modified_gmt":"2019-12-05T05:17:35","slug":"using-the-vertical-line-test","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/using-the-vertical-line-test\/","title":{"raw":"Identify a One-to-One Function","rendered":"Identify a One-to-One Function"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Define a one-to-one function<\/li>\n \t<li>Use the horizontal&nbsp;line test to determine whether a function is one-to-one<\/li>\n<\/ul>\n<\/div>\nRemember that in a function, the input value must have one and only one value for the output.&nbsp;There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the <b>domain of the function<\/b>. The set of output values is called the <b>range of the function<\/b>.\n\nIn the first example, we remind you how to define domain and range using a table of values.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nFind the domain and range for the function.\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<th style=\"text-align: center;\"><strong><i>x<\/i><\/strong><\/th>\n<th style=\"text-align: center;\"><strong><i>y<\/i><\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22125[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22126[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[reveal-answer q=\"130987\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"130987\"]\n\nThe domain is the set of inputs or <i>x<\/i>-coordinates.\n<p align=\"center\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\nThe range is the set of outputs or&nbsp;<i>y<\/i>-coordinates.\n<p align=\"center\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\n<p align=\"center\">[\/hidden-answer]<\/p>\n\n<\/div>\nIn the following video, we show another example of finding domain and range from tabular data.\n\nhttps:\/\/youtu.be\/GPBq18fCEv4\n\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&nbsp;[latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of&nbsp;[latex]$1000[\/latex].\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\">\n<p id=\"fs-id1165135678633\">However, some functions have only one input value for each output value as well as having only one output value for each input value. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents as listed below.<\/p>\n\n<table style=\"width: 20%;\" summary=\"Two columns and five rows. The first column is labeled,\"><colgroup> <col> <col><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Letter grade<\/th>\n<th style=\"text-align: center;\">Grade point average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">A<\/td>\n<td style=\"text-align: center;\">[latex]4.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">B<\/td>\n<td style=\"text-align: center;\">[latex]3.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">C<\/td>\n<td style=\"text-align: center;\">[latex]2.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">D<\/td>\n<td style=\"text-align: center;\">[latex]1.0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\nTo visualize this concept, let us look again at the two simple functions sketched in (a) and (b) below. Note that (c) is not a function since the input&nbsp;<em>q<\/em> produces two outputs,&nbsp;<em>y<\/em> and&nbsp;<em>z<\/em>.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\">\n\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.\n<div class=\"textbox\">\n<h3>A General Note: One-to-One Function<\/h3>\nA one-to-one function is a function in which each output value corresponds to exactly one input value.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nDetermine whether each of the following tables represents a one-to-one function.\n\na)\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]12[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]-5[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nb)\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]32[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]32[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]64[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]64[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]128[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;\n\n[reveal-answer q=\"945171\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"945171\"]\n\nTable a) maps the output value&nbsp;[latex]2[\/latex] to two different input values, therefore&nbsp;this is NOT a one-to-one function.\n\nTable b) maps each output to one unique input, therefore this IS a one-to-one function.\n\nOnly table b) is one-to-one.\n\n[\/hidden-answer]\n\n<\/div>\nIn the following video, we show an example of using tables of values to determine whether a function is one-to-one.\n\nhttps:\/\/youtu.be\/QFOJmevha_Y\n<h2 style=\"text-align: left;\">Using the Horizontal Line Test<\/h2>\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function&nbsp;is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. &nbsp;To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\n\n<div class=\"note precalculus howto textbox\">\n<h3 style=\"text-align: left;\"><strong>How To: Given a graph of a function, use the horizontal line test&nbsp;<\/strong><strong>to determine if the graph represents a one-to-one function<\/strong><\/h3>\n<ol id=\"fs-id1165137611853\">\n \t<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\n \t<li>If there is any such line, determine that the function is not one-to-one.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\nFor&nbsp;the following graphs, determine which represent one-to-one functions.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\">\n[reveal-answer q=\"783411\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"783411\"]\n<p id=\"fs-id1165135185190\">The function in (a) is&nbsp;not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"294\" height=\"267\">\n<figure id=\"Figure_01_01_010\" class=\"small\"><\/figure>\nThe function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.\n\n<img class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\">\n\n&nbsp;\n\nThe function (c) is not one-to-one and is in fact not a function.\n\n<img class=\" wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\">\n\n&nbsp;\n\n[\/hidden-answer]\n\n<\/div>\nThe following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.\n\n[embed]https:\/\/youtu.be\/tbSGdcSN8RE[\/embed]\n<h2>Summary<\/h2>\nIn real life and in algebra, different variables are often linked. When a change in value of one variable causes a change in the value of another variable, their interaction is called a relation. A relation has an input value which corresponds to an output value. When each input value has one and only one output value, the relation is a function. When each output value has one and only one input value, the function is one-to-one. Functions can be written as ordered pairs, tables, or graphs. The set of input values is called the domain, and the set of output values is called the range.\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define a one-to-one function<\/li>\n<li>Use the horizontal&nbsp;line test to determine whether a function is one-to-one<\/li>\n<\/ul>\n<\/div>\n<p>Remember that in a function, the input value must have one and only one value for the output.&nbsp;There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the <b>domain of the function<\/b>. The set of output values is called the <b>range of the function<\/b>.<\/p>\n<p>In the first example, we remind you how to define domain and range using a table of values.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the domain and range for the function.<\/p>\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<th style=\"text-align: center;\"><strong><i>x<\/i><\/strong><\/th>\n<th style=\"text-align: center;\"><strong><i>y<\/i><\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22125[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22126[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]15[\/latex]<\/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=\"q130987\">Show Solution<\/span><\/p>\n<div id=\"q130987\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is the set of inputs or <i>x<\/i>-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\n<p>The range is the set of outputs or&nbsp;<i>y<\/i>-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\n<p style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of finding domain and range from tabular data.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Give the Domain and Range Given the Points in a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GPBq18fCEv4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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&nbsp;[latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of&nbsp;[latex]$1000[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/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 id=\"fs-id1165135678633\">However, some functions have only one input value for each output value as well as having only one output value for each input value. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents as listed below.<\/p>\n<table style=\"width: 20%;\" summary=\"Two columns and five rows. The first column is labeled,\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Letter grade<\/th>\n<th style=\"text-align: center;\">Grade point average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">A<\/td>\n<td style=\"text-align: center;\">[latex]4.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">B<\/td>\n<td style=\"text-align: center;\">[latex]3.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">C<\/td>\n<td style=\"text-align: center;\">[latex]2.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">D<\/td>\n<td style=\"text-align: center;\">[latex]1.0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n<p>To visualize this concept, let us look again at the two simple functions sketched in (a) and (b) below. Note that (c) is not a function since the input&nbsp;<em>q<\/em> produces two outputs,&nbsp;<em>y<\/em> and&nbsp;<em>z<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/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<\/h3>\n<p>Determine whether each of the following tables represents a one-to-one function.<\/p>\n<p>a)<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]12[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]-5[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>b)<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]32[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]32[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]64[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]64[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]128[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q945171\">Show Solution<\/span><\/p>\n<div id=\"q945171\" class=\"hidden-answer\" style=\"display: none\">\n<p>Table a) maps the output value&nbsp;[latex]2[\/latex] to two different input values, therefore&nbsp;this is NOT a one-to-one function.<\/p>\n<p>Table b) maps each output to one unique input, therefore this IS a one-to-one function.<\/p>\n<p>Only table b) is one-to-one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show an example of using tables of values to determine whether a function is one-to-one.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Determine if a Relation Given as a Table is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QFOJmevha_Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 style=\"text-align: left;\">Using the Horizontal Line Test<\/h2>\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function&nbsp;is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. &nbsp;To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\n<div class=\"note precalculus howto textbox\">\n<h3 style=\"text-align: left;\"><strong>How To: Given a graph of a function, use the horizontal line test&nbsp;<\/strong><strong>to determine if the graph represents a one-to-one function<\/strong><\/h3>\n<ol id=\"fs-id1165137611853\">\n<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, determine that the function is not one-to-one.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>For&nbsp;the following graphs, determine which represent one-to-one functions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783411\">Show Solution<\/span><\/p>\n<div id=\"q783411\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135185190\">The function in (a) is&nbsp;not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"294\" height=\"267\" \/><\/p>\n<figure id=\"Figure_01_01_010\" class=\"small\"><\/figure>\n<p>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The function (c) is not one-to-one and is in fact not a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Determine if the Graph of a Relation is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tbSGdcSN8RE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>In real life and in algebra, different variables are often linked. When a change in value of one variable causes a change in the value of another variable, their interaction is called a relation. A relation has an input value which corresponds to an output value. When each input value has one and only one output value, the relation is a function. When each output value has one and only one input value, the function is one-to-one. Functions can be written as ordered pairs, tables, or graphs. The set of input values is called the domain, and the set of output values is called the range.<\/p>\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-15965\">\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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Determine if a Relation Given as a Table is a One-to-One Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QFOJmevha_Y\">https:\/\/youtu.be\/QFOJmevha_Y<\/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>Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5Z8DaZPJLKY\">https:\/\/youtu.be\/5Z8DaZPJLKY<\/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>Unit 17: Functions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":23485,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Determine if a Relation Given as a Table is a One-to-One Function\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QFOJmevha_Y\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/5Z8DaZPJLKY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 17: Functions, from Developmental Math: An Open 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