{"id":125,"date":"2023-06-21T13:22:35","date_gmt":"2023-06-21T13:22:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/identify-functions-using-graphs\/"},"modified":"2023-08-21T21:49:11","modified_gmt":"2023-08-21T21:49:11","slug":"identify-functions-using-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/identify-functions-using-graphs\/","title":{"raw":"\u25aa   Identify Functions using Graphs","rendered":"\u25aa   Identify Functions using Graphs"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Verify a function using the vertical line test<\/li>\r\n \t<li>Verify a one-to-one function with the horizontal line test<\/li>\r\n \t<li>Identify the graphs of the toolkit functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis.\r\n\r\nThe most common graphs name the input value [latex]x[\/latex] and the output value [latex]y[\/latex], and we say [latex]y[\/latex] is a function of [latex]x[\/latex], or [latex]y=f\\left(x\\right)[\/latex] when the function is named [latex]f[\/latex]. The graph of the function is the set of all points [latex]\\left(x,y\\right)[\/latex] in the plane that satisfies the equation [latex]y=f\\left(x\\right)[\/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em>x<\/em>-coordinate of each point is an input value and the <em>y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in the graph below\u00a0tell us that [latex]f\\left(0\\right)=2[\/latex] and [latex]f\\left(6\\right)=1[\/latex]. However, the set of all points [latex]\\left(x,y\\right)[\/latex] satisfying [latex]y=f\\left(x\\right)[\/latex] is a curve. The curve shown includes [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex] because the curve passes through those points.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191012\/CNX_Precalc_Figure_01_01_0112.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" \/>\r\n\r\nThe <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. A vertical line includes all points with a particular [latex]x[\/latex] value. The [latex]y[\/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[\/latex] value. If we can draw <em>any<\/em> vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because that [latex]x[\/latex] value has more than one output. A function has only one output value for each input value.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191014\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a graph, use the vertical line test to determine if the graph represents a function.<\/h3>\r\n<ol>\r\n \t<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If there is any such line, the graph does not represent a function.<\/li>\r\n \t<li>If no vertical line can intersect the curve more than once, the graph does represent a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying the Vertical Line Test<\/h3>\r\nWhich of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/>\r\n[reveal-answer q=\"689864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"689864\"]\r\n\r\nIf any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191020\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDoes the graph below represent a function?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/>\r\n[reveal-answer q=\"783855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783855\"]\r\n\r\nYes.\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"750\"]40676[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/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=\"wp-image-6721 aligncenter\" 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=\"\" width=\"417\" height=\"395\" \/>\r\n\r\n[reveal-answer q=\"569537\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"569537\"]\r\n<ul>\r\n \t<li>Red: Not a function<\/li>\r\n \t<li>Orange: Function<\/li>\r\n \t<li>Blue: Function<\/li>\r\n \t<li>Green: Not a fucntion<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Horizontal Line Test<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>relations, functions, and one-to-one functions<\/h3>\r\nA relation is a set of ordered pairs, inputs and their corresponding outputs, [latex]\\left(x, y\\right)[\/latex].\r\n\r\nA function is a particular relation in which each input corresponds to exactly one output,\u00a0[latex]\\left(x, f(x)\\right)[\/latex].\r\n\r\nA one-to-one function is a particular function in which each output corresponds to exactly one input.\r\n\r\n<\/div>\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. A horizontal line includes all points with a particular [latex]y[\/latex] value. The [latex]x[\/latex] value of a point where a vertical line intersects a function represents the input for that output [latex]y[\/latex] value. 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<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=\"textbox exercises\">\r\n<h3>Example: Applying the Horizontal Line Test<\/h3>\r\nConsider the functions (a), and (b)shown in\u00a0the graphs\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\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/>\r\n\r\nAre either of the functions one-to-one?\r\n\r\n[reveal-answer q=\"173050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173050\"]\r\n\r\nThe 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<img class=\"aligncenter\" 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\" width=\"487\" height=\"445\" \/>\r\n\r\nThe function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"750\"]111715[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/tbSGdcSN8RE\r\n<h2>Identifying Basic Toolkit Functions<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>toolkit functions<\/h3>\r\nYou should memorize the shapes of these graphs along with their names in words and in equation form. Be able to plot a few points of each to the left and right of the y-axis. You'll be using these function families throughout the rest of the course.\r\n\r\n<\/div>\r\nIn this text we explore functions\u2014the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our \"toolkit functions,\" which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[\/latex] as the input variable and [latex]y=f\\left(x\\right)[\/latex] as the output variable.\r\n\r\nWe will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th colspan=\"3\">Toolkit Functions<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Name<\/th>\r\n<th>Function<\/th>\r\n<th>Graph<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Constant<\/td>\r\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191028\/CNX_Precalc_Figure_01_01_018n.jpg\" alt=\"Graph of a constant function.\" width=\"517\" height=\"319\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identity<\/td>\r\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191030\/CNX_Precalc_Figure_01_01_019n.jpg\" alt=\"Graph of a straight line.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Absolute value<\/td>\r\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191034\/CNX_Precalc_Figure_01_01_020n.jpg\" alt=\"Graph of absolute function.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quadratic<\/td>\r\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191037\/CNX_Precalc_Figure_01_01_021n.jpg\" alt=\"Graph of a parabola.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cubic<\/td>\r\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191039\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reciprocal\/ Rational<\/td>\r\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191042\/CNX_Precalc_Figure_01_01_023n.jpg\" alt=\"Graph of f(x)=1\/x.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reciprocal \/ Rational squared<\/td>\r\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191044\/CNX_Precalc_Figure_01_01_024n.jpg\" alt=\"Graph of f(x)=1\/x^2.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Square root<\/td>\r\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191047\/CNX_Precalc_Figure_01_01_025n.jpg\" alt=\"Graph of f(x)=sqrt(x).\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cube root<\/td>\r\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191050\/CNX_Precalc_Figure_01_01_026n.jpg\" alt=\"Graph of f(x)=x^(1\/3).\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"780\"]111722[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIn this exercise, you will reference the nine toolkit functions presented above.\r\n<ol>\r\n \t<li>Graph each toolkit function using function notation on an online graphing calculator.<\/li>\r\n \t<li>Make a table of values that references the function and includes at least the interval [-5,5].<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Verify a function using the vertical line test<\/li>\n<li>Verify a one-to-one function with the horizontal line test<\/li>\n<li>Identify the graphs of the toolkit functions<\/li>\n<\/ul>\n<\/div>\n<p>As we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\n<p>The most common graphs name the input value [latex]x[\/latex] and the output value [latex]y[\/latex], and we say [latex]y[\/latex] is a function of [latex]x[\/latex], or [latex]y=f\\left(x\\right)[\/latex] when the function is named [latex]f[\/latex]. The graph of the function is the set of all points [latex]\\left(x,y\\right)[\/latex] in the plane that satisfies the equation [latex]y=f\\left(x\\right)[\/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em>x<\/em>-coordinate of each point is an input value and the <em>y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in the graph below\u00a0tell us that [latex]f\\left(0\\right)=2[\/latex] and [latex]f\\left(6\\right)=1[\/latex]. However, the set of all points [latex]\\left(x,y\\right)[\/latex] satisfying [latex]y=f\\left(x\\right)[\/latex] is a curve. The curve shown includes [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex] because the curve passes through those points.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191012\/CNX_Precalc_Figure_01_01_0112.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"442\" \/><\/p>\n<p>The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. A vertical line includes all points with a particular [latex]x[\/latex] value. The [latex]y[\/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[\/latex] value. If we can draw <em>any<\/em> vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because that [latex]x[\/latex] value has more than one output. A function has only one output value for each input value.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191014\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given a graph, use the vertical line test to determine if the graph represents a function.<\/h3>\n<ol>\n<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, the graph does not represent a function.<\/li>\n<li>If no vertical line can intersect the curve more than once, the graph does represent a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying the Vertical Line Test<\/h3>\n<p>Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q689864\">Show Solution<\/span><\/p>\n<div id=\"q689864\" class=\"hidden-answer\" style=\"display: none\">\n<p>If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191020\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Does the graph below represent a function?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191023\/CNX_Precalc_Figure_01_01_017.jpg\" alt=\"Graph of absolute value function.\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783855\">Show Solution<\/span><\/p>\n<div id=\"q783855\" class=\"hidden-answer\" style=\"display: none\">\n<p>Yes.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm40676\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40676&theme=oea&iframe_resize_id=ohm40676&show_question_numbers\" width=\"100%\" height=\"750\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/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=\"wp-image-6721 aligncenter\" 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=\"\" width=\"417\" height=\"395\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q569537\">Show Answer<\/span><\/p>\n<div id=\"q569537\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Red: Not a function<\/li>\n<li>Orange: Function<\/li>\n<li>Blue: Function<\/li>\n<li>Green: Not a fucntion<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Horizontal Line Test<\/h2>\n<div class=\"textbox examples\">\n<h3>relations, functions, and one-to-one functions<\/h3>\n<p>A relation is a set of ordered pairs, inputs and their corresponding outputs, [latex]\\left(x, y\\right)[\/latex].<\/p>\n<p>A function is a particular relation in which each input corresponds to exactly one output,\u00a0[latex]\\left(x, f(x)\\right)[\/latex].<\/p>\n<p>A one-to-one function is a particular function in which each output corresponds to exactly one input.<\/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 <strong>horizontal line test<\/strong>. Draw horizontal lines through the graph. A horizontal line includes all points with a particular [latex]y[\/latex] value. The [latex]x[\/latex] value of a point where a vertical line intersects a function represents the input for that output [latex]y[\/latex] value. 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<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=\"textbox exercises\">\n<h3>Example: Applying the Horizontal Line Test<\/h3>\n<p>Consider the functions (a), and (b)shown in\u00a0the graphs\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\/18191017\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<p>Are either of the functions one-to-one?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q173050\">Show Solution<\/span><\/p>\n<div id=\"q173050\" class=\"hidden-answer\" style=\"display: none\">\n<p>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><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" width=\"487\" height=\"445\" alt=\"image\" \/><\/p>\n<p>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm111715\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111715&theme=oea&iframe_resize_id=ohm111715&show_question_numbers\" width=\"100%\" height=\"750\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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>Identifying Basic Toolkit Functions<\/h2>\n<div class=\"textbox examples\">\n<h3>toolkit functions<\/h3>\n<p>You should memorize the shapes of these graphs along with their names in words and in equation form. Be able to plot a few points of each to the left and right of the y-axis. You&#8217;ll be using these function families throughout the rest of the course.<\/p>\n<\/div>\n<p>In this text we explore functions\u2014the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our &#8220;toolkit functions,&#8221; which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[\/latex] as the input variable and [latex]y=f\\left(x\\right)[\/latex] as the output variable.<\/p>\n<p>We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"3\">Toolkit Functions<\/th>\n<\/tr>\n<tr>\n<th>Name<\/th>\n<th>Function<\/th>\n<th>Graph<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Constant<\/td>\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191028\/CNX_Precalc_Figure_01_01_018n.jpg\" alt=\"Graph of a constant function.\" width=\"517\" height=\"319\" \/><\/td>\n<\/tr>\n<tr>\n<td>Identity<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191030\/CNX_Precalc_Figure_01_01_019n.jpg\" alt=\"Graph of a straight line.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Absolute value<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191034\/CNX_Precalc_Figure_01_01_020n.jpg\" alt=\"Graph of absolute function.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Quadratic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191037\/CNX_Precalc_Figure_01_01_021n.jpg\" alt=\"Graph of a parabola.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Cubic<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191039\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Reciprocal\/ Rational<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191042\/CNX_Precalc_Figure_01_01_023n.jpg\" alt=\"Graph of f(x)=1\/x.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Reciprocal \/ Rational squared<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191044\/CNX_Precalc_Figure_01_01_024n.jpg\" alt=\"Graph of f(x)=1\/x^2.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Square root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191047\/CNX_Precalc_Figure_01_01_025n.jpg\" alt=\"Graph of f(x)=sqrt(x).\" \/><\/td>\n<\/tr>\n<tr>\n<td>Cube root<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191050\/CNX_Precalc_Figure_01_01_026n.jpg\" alt=\"Graph of f(x)=x^(1\/3).\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm111722\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111722&theme=oea&iframe_resize_id=ohm111722&show_question_numbers\" width=\"100%\" height=\"780\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>In this exercise, you will reference the nine toolkit functions presented above.<\/p>\n<ol>\n<li>Graph each toolkit function using function notation on an online graphing calculator.<\/li>\n<li>Make a table of values that references the function and includes at least the interval [-5,5].<\/li>\n<\/ol>\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-125\">\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>Question ID 111715, 11722. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>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>Vertical Line Test. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/dcq8twow2q\">https:\/\/www.desmos.com\/calculator\/dcq8twow2q<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 40676. <strong>Authored by<\/strong>: Jenck, Michael. <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>\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":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 111715, 11722\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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