{"id":119,"date":"2023-06-21T13:22:35","date_gmt":"2023-06-21T13:22:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/function-notation-and-graphs-of-functions\/"},"modified":"2023-07-04T03:07:32","modified_gmt":"2023-07-04T03:07:32","slug":"function-notation-and-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/function-notation-and-graphs-of-functions\/","title":{"raw":"\u25aa   Function Notation and Graphs of Functions","rendered":"\u25aa   Function Notation and Graphs of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write functions using algebraic notation<\/li>\r\n \t<li>Use the vertical line test to determine whether a graph represents a function<\/li>\r\n<\/ul>\r\n<\/div>\r\nSome people think of functions as \u201cmathematical machines.\u201d Imagine you have a machine that changes a number according to a specific rule such as \u201cmultiply by\u00a0[latex]3[\/latex] and add\u00a0[latex]2[\/latex]\u201d or \u201cdivide by\u00a0[latex]5[\/latex], add\u00a0[latex]25[\/latex], and multiply by [latex]\u22121[\/latex].\u201d If you put a number into the machine, a new number will pop out the other end having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.\r\n\r\nYou can also call the machine \u201c<i>f\u201d <\/i>for function. If you put <i>x <\/i>into the box, <i>f<\/i>(<i>x<\/i>)<i>, <\/i>comes out. Mathematically speaking, <i>x<\/i> is the input, or the \u201cindependent variable,\u201d and <i>f<\/i>(<i>x<\/i>) is the output, or the \u201cdependent variable,\u201d since it depends on the value of <i>x<\/i>.\r\n\r\n[latex]f(x)=4x+1[\/latex] is written in function notation and is read \u201c<i>f <\/i>of <i>x<\/i> equals\u00a0[latex]4x[\/latex] plus\u00a0[latex]1[\/latex].\u201d It represents the following situation: A function named <i>f <\/i>acts upon an input, <i>x, <\/i>and produces <i>f<\/i>(<i>x<\/i>) which is equal to [latex]4x+1[\/latex]. This is the same as the equation [latex]y=4x+1[\/latex].\r\n\r\nFunction notation gives you more flexibility because you do not have to use <i>y<\/i> for every equation. Instead, you could use <i>f<\/i>(<i>x<\/i>) or <i>g<\/i>(<i>x<\/i>) or <i>c<\/i>(<i>x<\/i>). This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.\r\n<h2>Using Function Notation<\/h2>\r\nOnce we determine that a relationship is a function, we need to display and define the functional relationship so that we can understand it, use it, and possibly even program it into a computer. There are various ways of representing functions. A standard <strong>function notation<\/strong> is one representation that facilitates working with functions.\r\n\r\nNow you try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRepresent height as a function of age using function notation.\r\n[reveal-answer q=\"221008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"221008\"]\r\n\r\nTo represent \"height as a function of age,\" we start by identifying the variables: [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age.\r\n\r\n[latex]\\begin{array}{ccc}h\\text{ is }f\\text{ of }a\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We name the function }f;\\text{ height is a function of age}.\\hfill \\\\ h=f\\left(a\\right)\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We use parentheses to indicate the function input}\\text{. }\\hfill \\\\ f\\left(a\\right)\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We name the function }f;\\text{ the expression is read as ''}f\\text{ of }a\\text{.''}\\hfill \\end{array}[\/latex]\r\n\r\nNote:\u00a0We can use any letter to name the function; the notation [latex]h=f\\left(a\\right)[\/latex] shows us that [latex]h[\/latex] depends on, or is a function of, [latex]a[\/latex]. The value [latex]a[\/latex] must be put into the function [latex]f[\/latex] to get a result (height). The parentheses indicate that age is the input for the function; they do not indicate multiplication.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow try another.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<ol>\r\n \t<li>Write the formula for perimeter of a square, [latex]P=4s[\/latex], as a function.<\/li>\r\n \t<li>Write the formula for area of a square, [latex]A=l^{2}[\/latex], as a function.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"136183\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"136183\"]\r\n<ol>\r\n \t<li>Name the function P. P is a function of the length of the sides, s. Perimeter, as a function of side length, is equal to\u00a0[latex]4[\/latex] times side length.\u00a0[latex]P(s)=4s[\/latex]<\/li>\r\n \t<li>Name the function A. \u00a0Area, as a function of the length of the sides, is equal to the length squared.[latex]A(l)=l^{2}[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis would make it easy to graph both functions on the same graph without confusion about the variables.\r\n\r\nWe can also use an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means \"first add <em>a<\/em> and <em>b<\/em>, and the result is the input for the function <em>f<\/em>.\" The operations must be performed in this order to obtain the correct result.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Function Notation<\/h3>\r\nThe notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as \"[latex]y[\/latex] is a function of [latex]x[\/latex].\" The letter [latex]x[\/latex] represents the input value, or independent variable. The letter <em>[latex]y[\/latex] <\/em>or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse function notation to represent a function whose input is the name of a month and output is the number of days in that month.\r\n\r\n[reveal-answer q=\"5489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5489\"]\r\n\r\nThe number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a \"rule\" that associates a specific number (the output) with each input.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200459\/CNX_Precalc_Figure_01_01_0052.jpg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" \/>\r\n\r\nFor example, [latex]f\\left(\\text{March}\\right)=31[\/latex], because March has\u00a0[latex]31[\/latex] days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?\r\n\r\n[reveal-answer q=\"226737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"226737\"]\r\n\r\nWhen we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers, [latex]N[\/latex], is 300. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were\u00a0[latex]300[\/latex] police officers in the town.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following videos we show two more\u00a0examples of how to express a relationship using function notation.\r\n\r\nhttps:\/\/youtu.be\/lF0fzdaxU_8\r\n\r\nhttps:\/\/youtu.be\/nAF_GZFwU1g\r\n<h2>Graphs of Functions<\/h2>\r\nWhen both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the <i>x<\/i>-axis and the dependent value is plotted on the <i>y<\/i>-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as\u00a0[latex]y = f(x)[\/latex], or y is a function of\u00a0[latex]x[\/latex], we would write ordered pairs\u00a0[latex](x, f(x))[\/latex] using function notation instead of\u00a0[latex](x,y)[\/latex] as you may have seen previously.\r\n\r\n<img class=\"wp-image-2679 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16195741\/Screen-Shot-2016-07-16-at-12.56.58-PM-300x281.png\" alt=\"Coordinate axes with y-axis labeled f(x) and the x axis labeled x.\" width=\"408\" height=\"382\" \/>\r\n\r\nWe can identify whether the graph of a relation represents a function\u00a0because for each <i>x<\/i>-coordinate there will be exactly one <i>y<\/i>-coordinate.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/><\/p>\r\nWhen a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of <i>x<\/i>.\r\n\r\nIf, on the other hand, a graph shows two or more intersections with a vertical line, then an input (<i>x<\/i>-coordinate) can have more than one output (<i>y<\/i>-coordinate), and <i>y<\/i> is not a function of <i>x<\/i>. Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the \u201cvertical line test.\u201d\r\n\r\nYou try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the vertical line test to determine whether the relation plotted on this graph is a function.\r\n\r\n<img class=\"size-medium wp-image-2680 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16200323\/Screen-Shot-2016-07-16-at-1.02.50-PM-300x275.png\" alt=\"Graph with circle plotted - center at (0,0) radius = 2, more points include (2,0), (-2,0)\" width=\"300\" height=\"275\" \/>\r\n[reveal-answer q=\"28965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"28965\"]\r\n\r\nThis relationship cannot be a function, because some of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232418\/image002.jpg\" alt=\"A circle with four vertical lines through it. Three of the lines cross the circle at two points, and one line crosses the edge of the circle at one point.\" width=\"340\" height=\"343\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe vertical line method can also be applied to a set of ordered pairs plotted on a coordinate plane to determine if the relation is a function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the ordered pairs [latex]\\{(\u22121,3),(\u22122,5),(\u22123,3),(\u22125,\u22123)\\}[\/latex] plotted on the graph below. Use the vertical line test to determine whether the set of ordered pairs represents a function.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232420\/image003.jpg\" alt=\"The points (\u22121,3); (\u22122,5); (\u22123,3); and (\u22125,\u22123).\" width=\"329\" height=\"328\" \/><\/p>\r\n<p align=\"center\">[reveal-answer q=\"114452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"114452\"]<\/p>\r\n<p align=\"center\">Drawing vertical lines through each point results in each line only touching one point. This means that none of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates, so this is a function.<\/p>\r\n<p align=\"center\"><img class=\"size-full wp-image-2514 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/14213709\/Edit_Page_%E2%80%B9_Intermediate_Algebra_%E2%80%94_Pressbooks.png\" alt=\"Edit_Page_\u2039_Intermediate_Algebra_\u2014_Pressbooks\" width=\"298\" height=\"294\" \/><\/p>\r\n<p align=\"center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nNow consider the set of ordered pairs [latex]\\{(3,\u22121),(5,\u22122),(3,\u22123),(\u22123,5)\\}[\/latex]. One of the inputs,\u00a0[latex]3[\/latex], can produce two different outputs, [latex]\u22121[\/latex] and [latex]\u22123[\/latex]. You know what that means\u2014this set of ordered pairs is not a function. A plot confirms this.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232421\/image004.jpg\" alt=\"The point negative 3, 5; the point 5, negative 2. A line through the points negative 1, 3; and the point negative 3, negative 3.\" width=\"337\" height=\"336\" \/><\/p>\r\nNotice that a vertical line passes through two plotted points. One <i>x<\/i>-coordinate has multiple <i>y<\/i>-coordinates. This relation is not a function.\r\n\r\nIn the following video, we show another example of determining whether a graph represents a function using the vertical line test.\r\n\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write functions using algebraic notation<\/li>\n<li>Use the vertical line test to determine whether a graph represents a function<\/li>\n<\/ul>\n<\/div>\n<p>Some people think of functions as \u201cmathematical machines.\u201d Imagine you have a machine that changes a number according to a specific rule such as \u201cmultiply by\u00a0[latex]3[\/latex] and add\u00a0[latex]2[\/latex]\u201d or \u201cdivide by\u00a0[latex]5[\/latex], add\u00a0[latex]25[\/latex], and multiply by [latex]\u22121[\/latex].\u201d If you put a number into the machine, a new number will pop out the other end having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.<\/p>\n<p>You can also call the machine \u201c<i>f\u201d <\/i>for function. If you put <i>x <\/i>into the box, <i>f<\/i>(<i>x<\/i>)<i>, <\/i>comes out. Mathematically speaking, <i>x<\/i> is the input, or the \u201cindependent variable,\u201d and <i>f<\/i>(<i>x<\/i>) is the output, or the \u201cdependent variable,\u201d since it depends on the value of <i>x<\/i>.<\/p>\n<p>[latex]f(x)=4x+1[\/latex] is written in function notation and is read \u201c<i>f <\/i>of <i>x<\/i> equals\u00a0[latex]4x[\/latex] plus\u00a0[latex]1[\/latex].\u201d It represents the following situation: A function named <i>f <\/i>acts upon an input, <i>x, <\/i>and produces <i>f<\/i>(<i>x<\/i>) which is equal to [latex]4x+1[\/latex]. This is the same as the equation [latex]y=4x+1[\/latex].<\/p>\n<p>Function notation gives you more flexibility because you do not have to use <i>y<\/i> for every equation. Instead, you could use <i>f<\/i>(<i>x<\/i>) or <i>g<\/i>(<i>x<\/i>) or <i>c<\/i>(<i>x<\/i>). This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.<\/p>\n<h2>Using Function Notation<\/h2>\n<p>Once we determine that a relationship is a function, we need to display and define the functional relationship so that we can understand it, use it, and possibly even program it into a computer. There are various ways of representing functions. A standard <strong>function notation<\/strong> is one representation that facilitates working with functions.<\/p>\n<p>Now you try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Represent height as a function of age using function notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q221008\">Show Solution<\/span><\/p>\n<div id=\"q221008\" class=\"hidden-answer\" style=\"display: none\">\n<p>To represent &#8220;height as a function of age,&#8221; we start by identifying the variables: [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age.<\/p>\n<p>[latex]\\begin{array}{ccc}h\\text{ is }f\\text{ of }a\\hfill & \\hfill & \\hfill & \\hfill & \\text{We name the function }f;\\text{ height is a function of age}.\\hfill \\\\ h=f\\left(a\\right)\\hfill & \\hfill & \\hfill & \\hfill & \\text{We use parentheses to indicate the function input}\\text{. }\\hfill \\\\ f\\left(a\\right)\\hfill & \\hfill & \\hfill & \\hfill & \\text{We name the function }f;\\text{ the expression is read as ''}f\\text{ of }a\\text{.''}\\hfill \\end{array}[\/latex]<\/p>\n<p>Note:\u00a0We can use any letter to name the function; the notation [latex]h=f\\left(a\\right)[\/latex] shows us that [latex]h[\/latex] depends on, or is a function of, [latex]a[\/latex]. The value [latex]a[\/latex] must be put into the function [latex]f[\/latex] to get a result (height). The parentheses indicate that age is the input for the function; they do not indicate multiplication.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now try another.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<ol>\n<li>Write the formula for perimeter of a square, [latex]P=4s[\/latex], as a function.<\/li>\n<li>Write the formula for area of a square, [latex]A=l^{2}[\/latex], as a function.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q136183\">Show Solution<\/span><\/p>\n<div id=\"q136183\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Name the function P. P is a function of the length of the sides, s. Perimeter, as a function of side length, is equal to\u00a0[latex]4[\/latex] times side length.\u00a0[latex]P(s)=4s[\/latex]<\/li>\n<li>Name the function A. \u00a0Area, as a function of the length of the sides, is equal to the length squared.[latex]A(l)=l^{2}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>This would make it easy to graph both functions on the same graph without confusion about the variables.<\/p>\n<p>We can also use an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means &#8220;first add <em>a<\/em> and <em>b<\/em>, and the result is the input for the function <em>f<\/em>.&#8221; The operations must be performed in this order to obtain the correct result.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Function Notation<\/h3>\n<p>The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as &#8220;[latex]y[\/latex] is a function of [latex]x[\/latex].&#8221; The letter [latex]x[\/latex] represents the input value, or independent variable. The letter <em>[latex]y[\/latex] <\/em>or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q5489\">Show Solution<\/span><\/p>\n<div id=\"q5489\" class=\"hidden-answer\" style=\"display: none\">\n<p>The number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a &#8220;rule&#8221; that associates a specific number (the output) with each input.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200459\/CNX_Precalc_Figure_01_01_0052.jpg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" \/><\/p>\n<p>For example, [latex]f\\left(\\text{March}\\right)=31[\/latex], because March has\u00a0[latex]31[\/latex] days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q226737\">Show Solution<\/span><\/p>\n<div id=\"q226737\" class=\"hidden-answer\" style=\"display: none\">\n<p>When we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers, [latex]N[\/latex], is 300. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were\u00a0[latex]300[\/latex] police officers in the town.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following videos we show two more\u00a0examples of how to express a relationship using function notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Function Notation Application Problem\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lF0fzdaxU_8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Function Notation Application\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/nAF_GZFwU1g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Graphs of Functions<\/h2>\n<p>When both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the <i>x<\/i>-axis and the dependent value is plotted on the <i>y<\/i>-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as\u00a0[latex]y = f(x)[\/latex], or y is a function of\u00a0[latex]x[\/latex], we would write ordered pairs\u00a0[latex](x, f(x))[\/latex] using function notation instead of\u00a0[latex](x,y)[\/latex] as you may have seen previously.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2679 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16195741\/Screen-Shot-2016-07-16-at-12.56.58-PM-300x281.png\" alt=\"Coordinate axes with y-axis labeled f(x) and the x axis labeled x.\" width=\"408\" height=\"382\" \/><\/p>\n<p>We can identify whether the graph of a relation represents a function\u00a0because for each <i>x<\/i>-coordinate there will be exactly one <i>y<\/i>-coordinate.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/><\/p>\n<p>When a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of <i>x<\/i>.<\/p>\n<p>If, on the other hand, a graph shows two or more intersections with a vertical line, then an input (<i>x<\/i>-coordinate) can have more than one output (<i>y<\/i>-coordinate), and <i>y<\/i> is not a function of <i>x<\/i>. Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the \u201cvertical line test.\u201d<\/p>\n<p>You try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the vertical line test to determine whether the relation plotted on this graph is a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2680 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16200323\/Screen-Shot-2016-07-16-at-1.02.50-PM-300x275.png\" alt=\"Graph with circle plotted - center at (0,0) radius = 2, more points include (2,0), (-2,0)\" width=\"300\" height=\"275\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q28965\">Show Solution<\/span><\/p>\n<div id=\"q28965\" class=\"hidden-answer\" style=\"display: none\">\n<p>This relationship cannot be a function, because some of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232418\/image002.jpg\" alt=\"A circle with four vertical lines through it. Three of the lines cross the circle at two points, and one line crosses the edge of the circle at one point.\" width=\"340\" height=\"343\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The vertical line method can also be applied to a set of ordered pairs plotted on a coordinate plane to determine if the relation is a function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the ordered pairs [latex]\\{(\u22121,3),(\u22122,5),(\u22123,3),(\u22125,\u22123)\\}[\/latex] plotted on the graph below. Use the vertical line test to determine whether the set of ordered pairs represents a function.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232420\/image003.jpg\" alt=\"The points (\u22121,3); (\u22122,5); (\u22123,3); and (\u22125,\u22123).\" width=\"329\" height=\"328\" \/><\/p>\n<p style=\"text-align: center;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q114452\">Show Solution<\/span><\/p>\n<div id=\"q114452\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">Drawing vertical lines through each point results in each line only touching one point. This means that none of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates, so this is a function.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2514 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/14213709\/Edit_Page_%E2%80%B9_Intermediate_Algebra_%E2%80%94_Pressbooks.png\" alt=\"Edit_Page_\u2039_Intermediate_Algebra_\u2014_Pressbooks\" width=\"298\" height=\"294\" \/><\/p>\n<p style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<p>Now consider the set of ordered pairs [latex]\\{(3,\u22121),(5,\u22122),(3,\u22123),(\u22123,5)\\}[\/latex]. One of the inputs,\u00a0[latex]3[\/latex], can produce two different outputs, [latex]\u22121[\/latex] and [latex]\u22123[\/latex]. You know what that means\u2014this set of ordered pairs is not a function. A plot confirms this.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232421\/image004.jpg\" alt=\"The point negative 3, 5; the point 5, negative 2. A line through the points negative 1, 3; and the point negative 3, negative 3.\" width=\"337\" height=\"336\" \/><\/p>\n<p>Notice that a vertical line passes through two plotted points. One <i>x<\/i>-coordinate has multiple <i>y<\/i>-coordinates. This relation is not a function.<\/p>\n<p>In the following video, we show another example of determining whether a graph represents a function using the vertical line test.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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\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-119\">\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>Ex: Function Notation Application Problem. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lF0fzdaxU_8\">https:\/\/youtu.be\/lF0fzdaxU_8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Function Notation Application. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nAF_GZFwU1g\">https:\/\/youtu.be\/nAF_GZFwU1g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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@3.278:1\/Preface<\/li><li>Ex: Determine if a Table of Values Represents a Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/y2TqnP_6M1s\">https:\/\/youtu.be\/y2TqnP_6M1s<\/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":395986,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Function Notation Application Problem\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/lF0fzdaxU_8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Function Notation Application\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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