{"id":493,"date":"2025-07-10T17:44:45","date_gmt":"2025-07-10T17:44:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=493"},"modified":"2026-01-07T22:42:23","modified_gmt":"2026-01-07T22:42:23","slug":"working-with-functions-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/working-with-functions-background-youll-need-3\/","title":{"raw":"Working with Functions: Background You'll Need 3","rendered":"Working with Functions: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Determine if a function is one-to-one<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Determining Whether a Function is One-to-One<\/h2>\r\nSome functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown below, the stock price was [latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of [latex]$1000[\/latex].\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_9185\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-9185\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/16175429\/be97793ae3e427723869ee41e52cd09a3286f4c1.png\" alt=\"a graph of market prices.\" width=\"500\" height=\"305\" \/> Graph of market prices[\/caption]\r\n\r\n&nbsp;\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 the table below.\r\n<table style=\"width: 100%; text-align: center;\">\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 <strong>one-to-one function<\/strong>, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>one-to-one function<\/h3>\r\nA <strong>one-to-one function<\/strong> is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\">\r\n<ol>\r\n \t<li>Is a balance a one-to-one function of the bank account number?<\/li>\r\n \t<li>Is a bank account number a one-to-one 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] [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<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317892[\/ohm_question]<\/section>\r\n<h2>The Horizontal Line Test<\/h2>\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 <strong>horizontal line test<\/strong>. Draw horizontal lines through the graph. If we can draw <em>any<\/em> horizontal line that intersects a graph more than once, then the graph does <em>not<\/em> represent a one-to-one function because that [latex]y[\/latex] value has more than one input.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>horizontal line test<\/h3>\r\nThe <strong>horizontal line test<\/strong> checks if a function is one-to-one by ensuring that no horizontal line intersects the graph more than once.\r\n\r\n<\/section><section class=\"textbox questionHelp\"><strong>How To: Horizontal Line Test<\/strong>Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function following these steps.\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<\/section><section class=\"textbox example\">Which of the graphs represent(s) a one-to-one function?\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\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/> Graph of lines\/polynomials[\/caption]\r\n\r\n[reveal-answer q=\"173050\"]Show Solution[\/reveal-answer] [hidden-answer a=\"173050\"] The function in (a) 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.)\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img 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=\"\" width=\"487\" height=\"445\" \/> Graph of polynomial[\/caption]\r\n\r\nThe function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once. The function in (c) is not one-to-one. The horizontal line intersects the graph of the function at two points. [\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317893[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Determine if a function is one-to-one<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Determining Whether a Function is One-to-One<\/h2>\n<p>Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown below, the stock price was [latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of [latex]$1000[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_9185\" aria-describedby=\"caption-attachment-9185\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-9185\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/16175429\/be97793ae3e427723869ee41e52cd09a3286f4c1.png\" alt=\"a graph of market prices.\" width=\"500\" height=\"305\" \/><figcaption id=\"caption-attachment-9185\" class=\"wp-caption-text\">Graph of market prices<\/figcaption><\/figure>\n<p>&nbsp;<\/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 the table below.<\/p>\n<table style=\"width: 100%; text-align: center;\">\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 <strong>one-to-one function<\/strong>, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>one-to-one function<\/h3>\n<p>A <strong>one-to-one function<\/strong> is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<ol>\n<li>Is a balance a one-to-one function of the bank account number?<\/li>\n<li>Is a bank account number a one-to-one 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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q997233\">Show Solution<\/button> <\/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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317892\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317892&theme=lumen&iframe_resize_id=ohm317892&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>The Horizontal Line Test<\/h2>\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 <strong>horizontal line test<\/strong>. Draw horizontal lines through the graph. If we can draw <em>any<\/em> horizontal line that intersects a graph more than once, then the graph does <em>not<\/em> represent a one-to-one function because that [latex]y[\/latex] value has more than one input.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>horizontal line test<\/h3>\n<p>The <strong>horizontal line test<\/strong> checks if a function is one-to-one by ensuring that no horizontal line intersects the graph more than once.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\"><strong>How To: Horizontal Line Test<\/strong>Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function following these steps.<\/p>\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<\/section>\n<section class=\"textbox example\">Which of the graphs represent(s) a one-to-one function?<\/p>\n<figure style=\"width: 975px\" 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\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><figcaption class=\"wp-caption-text\">Graph of lines\/polynomials<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q173050\">Show Solution<\/button> <\/p>\n<div id=\"q173050\" class=\"hidden-answer\" style=\"display: none\"> The function in (a) 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.)<\/p>\n<p>&nbsp;<\/p>\n<figure style=\"width: 487px\" 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\/18191025\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"487\" height=\"445\" \/><figcaption class=\"wp-caption-text\">Graph of polynomial<\/figcaption><\/figure>\n<p>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once. The function in (c) is not one-to-one. The horizontal line intersects the graph of the function at two points. <\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317893\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317893&theme=lumen&iframe_resize_id=ohm317893&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":498,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/493"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/493\/revisions"}],"predecessor-version":[{"id":5230,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/493\/revisions\/5230"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/498"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/493\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=493"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=493"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=493"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}