{"id":863,"date":"2022-02-20T22:39:48","date_gmt":"2022-02-20T22:39:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=863"},"modified":"2026-04-18T00:18:52","modified_gmt":"2026-04-18T00:18:52","slug":"1-4-2-the-meanings-of-the-algebraic-forms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/1-4-2-the-meanings-of-the-algebraic-forms\/","title":{"raw":"1.4.2: The Meaning of the Algebraic Form of a Function","rendered":"1.4.2: The Meaning of the Algebraic Form of a Function"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Explain the meaning of assigning the independent variable of a function a value<\/li>\r\n \t<li>Determine function values given a domain value<\/li>\r\n \t<li>Explain the meaning of assigning a function a value<\/li>\r\n \t<li>Determine a domain value given a function value<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Assigning the Independent Variable a Value<\/h2>\r\n<h3>Determining function values<\/h3>\r\nThe algebraic form of a function defines the independent and dependent variables of the function. For example, the function [latex]f(x)=2x+1[\/latex], designates [latex]x[\/latex] as the the independent variable. We know this because [latex]x[\/latex] is the variable in parentheses next to the function name, [latex]f[\/latex]. The dependent variable is the function value, and is dependent on the value of the independent variable. The values of the independent variable determine the values of the function (the dependent variable) through the rule described in the algebraic form of the function.\r\n\r\nFor example, the function[latex]f(x)=2x+1[\/latex] has a function value of 1 when the independent variable [latex]x[\/latex] is 0. To determine this, we substitute [latex]x=0[\/latex] in the function:\r\n<p style=\"text-align: center;\">[latex]f(x)=2x+1\\\\f(0) = 2(0) + 1 =1[\/latex]<\/p>\r\nThe notation [latex]f(0)[\/latex] means the function value (or the value of the dependent variable) given the value of the independent variable [latex]x[\/latex] is 0. The function value is 1, after the independent variable in the rule is replaced with 0. Therefore, we may write [latex]f(0)=1[\/latex]. Similarly, the function value is 11 after the independent variable in the rule is replaced with 5.\r\n<p style=\"text-align: center;\">[latex] f(x)=2x+1\\\\f(5) = 2(5) + 1 = 10 + 1 = 11[\/latex]<\/p>\r\nOn the coordinate plane, a function is a graph. The graph is composed of those points [latex](x,y)[\/latex] where [latex]x[\/latex] is the independent variable and [latex]y=f(x)[\/latex] is the dependent variable. Therefore, when assigning the independent variable a value (e.g., [latex]x=5[\/latex]), it refers to the point on the graph where the [latex]x[\/latex]-coordinate is 5. According to the graph in figure 1, we can determine that the [latex]y[\/latex]-coordinate is 11 when the [latex]x[\/latex]-coordinate is 5 because the point (5, 11) lies on the graph of the function. The point (5, 11) is the point where the vertical line [latex]x=5[\/latex] intersects the graph [latex]y=f(x)[\/latex].\r\n\r\n[caption id=\"attachment_1749\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1749 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-300x300.png\" alt=\"Intersection of lines as described above. Choosing the vertical line x = 5 shows that f of 5 is 11\" width=\"300\" height=\"300\" \/> Figure 1.[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nDetermine the function values for [latex]g(x)=x^2+3x-2[\/latex]:\r\n<ol>\r\n \t<li>\u00a0[latex]g(0)[\/latex]<\/li>\r\n \t<li>\u00a0[latex]g(2)[\/latex]<\/li>\r\n \t<li>\u00a0[latex]g(\u20131)[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nReplace [latex]x[\/latex] in the function with each of the values 0, 2, and \u20131. Inserting the [latex]x[\/latex]-values inside parentheses helps keep the order of operations in our minds.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">g(0)<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">g(2)<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">g(-1)<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">1.\r\n\r\n[latex]g(x)=x^2+3x-2\\\\g(0)=(0)^2+3(0)-2=-2[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">2.\r\n\r\n[latex]g(x)=x^2+3x-2\\\\g(2)=(2)^2+3(2)-2\\\\g(2)=4+6-2=8[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">3.\r\n\r\n[latex]g(x)=x^2+3x-2\\\\g(-1)=(-1)^2+3(-1)-2\\\\g(-1)=1-3-2=-4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine the function values for [latex]f(x)=\\frac{2x}{x-3}[\/latex]:\r\n<ol>\r\n \t<li>\u00a0[latex]f(0)[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(-2)[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(3)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm410\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm410\"]\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center; vertical-align: top;\">f(0)<\/th>\r\n<th style=\"width: 33.3333%; text-align: center; vertical-align: top;\">f(-2)<\/th>\r\n<th style=\"width: 33.3333%; text-align: center; vertical-align: top;\">f(3)<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">1.\r\n[latex]f(x)=\\frac{2x}{x-3}\\\\f(0)=\\frac{2(0)}{0-3}\\\\f(0)=\\frac{0}{-3}=0[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">2.\r\n[latex]f(x)=\\frac{2x}{x-3}\\\\f(-2)=\\frac{2(-2)}{-2-3}\\\\f(-2)=\\frac{-4}{-5}=\\frac{4}{5}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">3.\r\n[latex]f(x)=\\frac{2x}{x-3}\\\\f(3)=\\frac{2(3)}{3-3}\\\\f(3)=\\frac{6}{0}[\/latex] is undefined. There is no function value at [latex]x=3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhy do you think there is no function value at [latex]x=3[\/latex] for the function\u00a0[latex]f(x)=\\frac{2x}{x-3}[\/latex]? If we look at the graph, we may gain a better understanding. Figure 2 shows the graph of the rational function\u00a0[latex]f(x)=\\frac{2x}{x-3}[\/latex]. The reason that [latex]f(3)[\/latex] is undefined is because there is a vertical asymptote at [latex]x=3[\/latex]. A graph can never cross a vertical asymptote, so there is no function value at [latex]x=3[\/latex].\r\n\r\n[caption id=\"attachment_1044\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1044 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-300x246.png\" alt=\"Graph of f(x) = 2x over x-3 described above.\" width=\"300\" height=\"246\" \/> Figure 2.\u00a0[latex]f(x)=\\frac{2x}{x-3}[\/latex][\/caption]\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nUse the graph to determine the function values [latex]f(x)[\/latex] at the given [latex]x[\/latex]-values.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1750 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-300x300.png\" alt=\"exponential function where x = 0 intersects at a height of -3, x = 1 at a height of -2, and x = -1 at a height of negative seven halves.\" width=\"300\" height=\"300\" \/><\/p>\r\n\r\n<ol>\r\n \t<li>\u00a0[latex]x=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=1[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=-1[\/latex]<\/li>\r\n<\/ol>\r\n&nbsp;\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<ol>\r\n \t<li>\u00a0The line [latex]x=0[\/latex] intersects the graph at the point (0, \u20133) so [latex]f(0) = -3[\/latex].<\/li>\r\n \t<li>\u00a0The line [latex]x=1[\/latex] intersects the graph at the point (1, \u20132) so [latex]f(1) = -2[\/latex].<\/li>\r\n \t<li>\u00a0The line [latex]x=-1[\/latex] intersects the graph at the point (-1, \u20134.5) so [latex]f(-1) = -3.5[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nUse the graph to determine the function values [latex]g(x)[\/latex] at the given [latex]x[\/latex]-values.\r\n<ol>\r\n \t<li>\u00a0[latex]x=1[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=4[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=16[\/latex]<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-1051\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/10010117\/log-graph-280x300.png\" alt=\"Logarithmic graph where x = 1 intersects at a value of 0, x = 4 at a value of 4, and x=16 at a value of 8.\" width=\"409\" height=\"438\" \/>\r\n\r\n[reveal-answer q=\"hjm878\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm878\"]\r\n<ol>\r\n \t<li>\u00a0[latex]f(1)=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(4)=4[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(16)=8[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Assigning the Function a Value<\/h2>\r\n<h3>Determining the value of the independent variable<\/h3>\r\nOn the coordinate plane, when a function is assigned a value, it refers to the point on the graph where the\u00a0[latex]y[\/latex]-coordinate of the point is the assigned value. For example, [latex]f(x)=7[\/latex], refers to a point on the graph where the [latex]y[\/latex]-coordinate is 7. If we have the graph of the function we can find the corresponding [latex]x[\/latex]-value of the function by determining the [latex]x[\/latex]-coordinate of the point where [latex]y=7[\/latex]. The graph in figure 3 shows that the point (3, 7) lies on the graph of the function. Therefore, when [latex]y=7,\\; x=3[\/latex]. This means that the function value of 7 occurs when the independent variable value is 3. The point (3, 7) is the point where the horizontal line [latex]y=7[\/latex] intersects the graph [latex]y=f(x)[\/latex].\r\n\r\n[caption id=\"attachment_1751\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1751 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-300x300.png\" alt=\"line with the horizontal line y=7 intersecting at an x value of 3, as described above.\" width=\"300\" height=\"300\" \/> Figure 3. The [latex]x[\/latex]-coordinate is 3 when [latex]f(x)=7[\/latex][\/caption]We can also determine the [latex]x[\/latex]-value algebraically, by setting [latex]f(x)=7[\/latex] and solving the resulting equation for [latex]x[\/latex].\r\n\r\n[latex]f(x) = 2x + 1, \\;\\;7=2x+1,\\;\\; 6=2x, \\;\\;3=x[\/latex]\r\n\r\nThis equation solving process to determine the value of an independent variable given a function value will be discussed throughout this entire course.\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nUse the graph of the function [latex]y=f(x)[\/latex] to determine the corresponding [latex]x[\/latex]-values when:\r\n<ol>\r\n \t<li>\u00a0[latex]f(x)=8[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(x)=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(x)=6[\/latex]<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-1047\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/10002654\/fx-2x6-240x300.png\" alt=\"Graph of f(x) = -2x+6 where the line y=8 intersects at an x value of -1, y=0 at an x value of 3, and y=6 at an x value of 0.\" width=\"339\" height=\"424\" \/>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>\u00a0The point where the horizontal line [latex]y=8[\/latex] meets the graph is at the point (\u20131, 8). Therefore, [latex]x=-1[\/latex] when [latex]f(x)=8[\/latex].<\/li>\r\n \t<li>\u00a0The point where the horizontal line [latex]y=0[\/latex] (the [latex]x[\/latex]-axis) meets the graph is at the point (3, 0). Therefore, [latex]x=3[\/latex] when [latex]f(x)=0[\/latex].<\/li>\r\n \t<li>\u00a0The point where the horizontal line [latex]y=6[\/latex] meets the graph is at the point (0, 6). Therefore, [latex]x=0[\/latex] when [latex]f(x)=6[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nUse the graph of the function [latex]y=g(x)[\/latex] to determine the corresponding [latex]x[\/latex]-values when:\r\n<ol>\r\n \t<li>\u00a0[latex]g(x)=4[\/latex]<\/li>\r\n \t<li>\u00a0[latex]g(x)=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]g(x)=-4[\/latex]<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-1048\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/10003434\/gxx%5E3-2x-293x300.png\" alt=\"Graph of a cubic function where the line y=4 intersects at x=2, y=0 intersects at x= -1.5, 0, and 1.5, y=-4 intersects at x=-2\" width=\"368\" height=\"377\" \/>\r\n\r\n[reveal-answer q=\"hjm635\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm635\"]\r\n<ol>\r\n \t<li>\u00a0[latex]x=2[\/latex] when [latex]g(x)=4[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=-1.5, 0, 1.5[\/latex] when [latex]g(x)=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=-2[\/latex] when [latex]g(x)=-4[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nFor [latex]T(x)=-2x+7[\/latex],\u00a0determine the corresponding [latex]x[\/latex]-values when:\r\n<ol>\r\n \t<li>\u00a0[latex]T(x)=1[\/latex]<\/li>\r\n \t<li>\u00a0[latex]T(x)=-1[\/latex]<\/li>\r\n \t<li>\u00a0[latex]T(x)=9[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nSet\u00a0[latex]T(x)[\/latex] equal to the given values and solve the resulting equation for [latex]x[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">T(x)=1<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">T(x)=-1<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">T(x)=9<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">1.\r\n\r\n[latex]T(x)=-2x+7,\\;\\;1=-2x+7,\\;\\;-6=-2x,\\;\\;3=x[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">2.\r\n\r\n[latex]T(x)=-2x+7,\\;\\;-1=-2x+7,\\;\\;-8=-2x,\\;\\;4=x[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">\u00a03.\r\n\r\n[latex]T(x)=-2x+7,\\;\\;9=-2x+7,\\;\\;2=-2x,\\;\\;-1=x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nFor [latex]T(x)=3x-2[\/latex],\u00a0determine the corresponding [latex]x[\/latex]-values when:\r\n<ol>\r\n \t<li>\u00a0[latex]T(x)=7[\/latex]<\/li>\r\n \t<li>\u00a0[latex]T(x)=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]T(x)=-11[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm757\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm757\"]\r\n<ol>\r\n \t<li>\u00a0[latex]x=3[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=\\frac{2}{3}[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x=-3[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Explain the meaning of assigning the independent variable of a function a value<\/li>\n<li>Determine function values given a domain value<\/li>\n<li>Explain the meaning of assigning a function a value<\/li>\n<li>Determine a domain value given a function value<\/li>\n<\/ul>\n<\/div>\n<h2>Assigning the Independent Variable a Value<\/h2>\n<h3>Determining function values<\/h3>\n<p>The algebraic form of a function defines the independent and dependent variables of the function. For example, the function [latex]f(x)=2x+1[\/latex], designates [latex]x[\/latex] as the the independent variable. We know this because [latex]x[\/latex] is the variable in parentheses next to the function name, [latex]f[\/latex]. The dependent variable is the function value, and is dependent on the value of the independent variable. The values of the independent variable determine the values of the function (the dependent variable) through the rule described in the algebraic form of the function.<\/p>\n<p>For example, the function[latex]f(x)=2x+1[\/latex] has a function value of 1 when the independent variable [latex]x[\/latex] is 0. To determine this, we substitute [latex]x=0[\/latex] in the function:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=2x+1\\\\f(0) = 2(0) + 1 =1[\/latex]<\/p>\n<p>The notation [latex]f(0)[\/latex] means the function value (or the value of the dependent variable) given the value of the independent variable [latex]x[\/latex] is 0. The function value is 1, after the independent variable in the rule is replaced with 0. Therefore, we may write [latex]f(0)=1[\/latex]. Similarly, the function value is 11 after the independent variable in the rule is replaced with 5.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=2x+1\\\\f(5) = 2(5) + 1 = 10 + 1 = 11[\/latex]<\/p>\n<p>On the coordinate plane, a function is a graph. The graph is composed of those points [latex](x,y)[\/latex] where [latex]x[\/latex] is the independent variable and [latex]y=f(x)[\/latex] is the dependent variable. Therefore, when assigning the independent variable a value (e.g., [latex]x=5[\/latex]), it refers to the point on the graph where the [latex]x[\/latex]-coordinate is 5. According to the graph in figure 1, we can determine that the [latex]y[\/latex]-coordinate is 11 when the [latex]x[\/latex]-coordinate is 5 because the point (5, 11) lies on the graph of the function. The point (5, 11) is the point where the vertical line [latex]x=5[\/latex] intersects the graph [latex]y=f(x)[\/latex].<\/p>\n<div id=\"attachment_1749\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1749\" class=\"wp-image-1749 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-300x300.png\" alt=\"Intersection of lines as described above. Choosing the vertical line x = 5 shows that f of 5 is 11\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-60.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1749\" class=\"wp-caption-text\">Figure 1.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Determine the function values for [latex]g(x)=x^2+3x-2[\/latex]:<\/p>\n<ol>\n<li>\u00a0[latex]g(0)[\/latex]<\/li>\n<li>\u00a0[latex]g(2)[\/latex]<\/li>\n<li>\u00a0[latex]g(\u20131)[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>Replace [latex]x[\/latex] in the function with each of the values 0, 2, and \u20131. Inserting the [latex]x[\/latex]-values inside parentheses helps keep the order of operations in our minds.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">g(0)<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">g(2)<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">g(-1)<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">1.<\/p>\n<p>[latex]g(x)=x^2+3x-2\\\\g(0)=(0)^2+3(0)-2=-2[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">2.<\/p>\n<p>[latex]g(x)=x^2+3x-2\\\\g(2)=(2)^2+3(2)-2\\\\g(2)=4+6-2=8[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">3.<\/p>\n<p>[latex]g(x)=x^2+3x-2\\\\g(-1)=(-1)^2+3(-1)-2\\\\g(-1)=1-3-2=-4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine the function values for [latex]f(x)=\\frac{2x}{x-3}[\/latex]:<\/p>\n<ol>\n<li>\u00a0[latex]f(0)[\/latex]<\/li>\n<li>\u00a0[latex]f(-2)[\/latex]<\/li>\n<li>\u00a0[latex]f(3)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm410\">Show Answer<\/span><\/p>\n<div id=\"qhjm410\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center; vertical-align: top;\">f(0)<\/th>\n<th style=\"width: 33.3333%; text-align: center; vertical-align: top;\">f(-2)<\/th>\n<th style=\"width: 33.3333%; text-align: center; vertical-align: top;\">f(3)<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">1.<br \/>\n[latex]f(x)=\\frac{2x}{x-3}\\\\f(0)=\\frac{2(0)}{0-3}\\\\f(0)=\\frac{0}{-3}=0[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">2.<br \/>\n[latex]f(x)=\\frac{2x}{x-3}\\\\f(-2)=\\frac{2(-2)}{-2-3}\\\\f(-2)=\\frac{-4}{-5}=\\frac{4}{5}[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: left; vertical-align: top;\">3.<br \/>\n[latex]f(x)=\\frac{2x}{x-3}\\\\f(3)=\\frac{2(3)}{3-3}\\\\f(3)=\\frac{6}{0}[\/latex] is undefined. There is no function value at [latex]x=3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Why do you think there is no function value at [latex]x=3[\/latex] for the function\u00a0[latex]f(x)=\\frac{2x}{x-3}[\/latex]? If we look at the graph, we may gain a better understanding. Figure 2 shows the graph of the rational function\u00a0[latex]f(x)=\\frac{2x}{x-3}[\/latex]. The reason that [latex]f(3)[\/latex] is undefined is because there is a vertical asymptote at [latex]x=3[\/latex]. A graph can never cross a vertical asymptote, so there is no function value at [latex]x=3[\/latex].<\/p>\n<div id=\"attachment_1044\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1044\" class=\"wp-image-1044 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-300x246.png\" alt=\"Graph of f(x) = 2x over x-3 described above.\" width=\"300\" height=\"246\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-300x246.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-768x631.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-1024x841.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-65x53.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-225x185.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3-350x288.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/fx2xx-3.png 1792w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1044\" class=\"wp-caption-text\">Figure 2.\u00a0[latex]f(x)=\\frac{2x}{x-3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Use the graph to determine the function values [latex]f(x)[\/latex] at the given [latex]x[\/latex]-values.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1750 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-300x300.png\" alt=\"exponential function where x = 0 intersects at a height of -3, x = 1 at a height of -2, and x = -1 at a height of negative seven halves.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-61.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol>\n<li>\u00a0[latex]x=0[\/latex]<\/li>\n<li>\u00a0[latex]x=1[\/latex]<\/li>\n<li>\u00a0[latex]x=-1[\/latex]<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<ol>\n<li>\u00a0The line [latex]x=0[\/latex] intersects the graph at the point (0, \u20133) so [latex]f(0) = -3[\/latex].<\/li>\n<li>\u00a0The line [latex]x=1[\/latex] intersects the graph at the point (1, \u20132) so [latex]f(1) = -2[\/latex].<\/li>\n<li>\u00a0The line [latex]x=-1[\/latex] intersects the graph at the point (-1, \u20134.5) so [latex]f(-1) = -3.5[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Use the graph to determine the function values [latex]g(x)[\/latex] at the given [latex]x[\/latex]-values.<\/p>\n<ol>\n<li>\u00a0[latex]x=1[\/latex]<\/li>\n<li>\u00a0[latex]x=4[\/latex]<\/li>\n<li>\u00a0[latex]x=16[\/latex]<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1051\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/10010117\/log-graph-280x300.png\" alt=\"Logarithmic graph where x = 1 intersects at a value of 0, x = 4 at a value of 4, and x=16 at a value of 8.\" width=\"409\" height=\"438\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm878\">Show Answer<\/span><\/p>\n<div id=\"qhjm878\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a0[latex]f(1)=0[\/latex]<\/li>\n<li>\u00a0[latex]f(4)=4[\/latex]<\/li>\n<li>\u00a0[latex]f(16)=8[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Assigning the Function a Value<\/h2>\n<h3>Determining the value of the independent variable<\/h3>\n<p>On the coordinate plane, when a function is assigned a value, it refers to the point on the graph where the\u00a0[latex]y[\/latex]-coordinate of the point is the assigned value. For example, [latex]f(x)=7[\/latex], refers to a point on the graph where the [latex]y[\/latex]-coordinate is 7. If we have the graph of the function we can find the corresponding [latex]x[\/latex]-value of the function by determining the [latex]x[\/latex]-coordinate of the point where [latex]y=7[\/latex]. The graph in figure 3 shows that the point (3, 7) lies on the graph of the function. Therefore, when [latex]y=7,\\; x=3[\/latex]. This means that the function value of 7 occurs when the independent variable value is 3. The point (3, 7) is the point where the horizontal line [latex]y=7[\/latex] intersects the graph [latex]y=f(x)[\/latex].<\/p>\n<div id=\"attachment_1751\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1751\" class=\"wp-image-1751 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-300x300.png\" alt=\"line with the horizontal line y=7 intersecting at an x value of 3, as described above.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-63.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1751\" class=\"wp-caption-text\">Figure 3. The [latex]x[\/latex]-coordinate is 3 when [latex]f(x)=7[\/latex]<\/p>\n<\/div>\n<p>We can also determine the [latex]x[\/latex]-value algebraically, by setting [latex]f(x)=7[\/latex] and solving the resulting equation for [latex]x[\/latex].<\/p>\n<p>[latex]f(x) = 2x + 1, \\;\\;7=2x+1,\\;\\; 6=2x, \\;\\;3=x[\/latex]<\/p>\n<p>This equation solving process to determine the value of an independent variable given a function value will be discussed throughout this entire course.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Use the graph of the function [latex]y=f(x)[\/latex] to determine the corresponding [latex]x[\/latex]-values when:<\/p>\n<ol>\n<li>\u00a0[latex]f(x)=8[\/latex]<\/li>\n<li>\u00a0[latex]f(x)=0[\/latex]<\/li>\n<li>\u00a0[latex]f(x)=6[\/latex]<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1047\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/10002654\/fx-2x6-240x300.png\" alt=\"Graph of f(x) = -2x+6 where the line y=8 intersects at an x value of -1, y=0 at an x value of 3, and y=6 at an x value of 0.\" width=\"339\" height=\"424\" \/><\/p>\n<h4>Solution<\/h4>\n<ol>\n<li>\u00a0The point where the horizontal line [latex]y=8[\/latex] meets the graph is at the point (\u20131, 8). Therefore, [latex]x=-1[\/latex] when [latex]f(x)=8[\/latex].<\/li>\n<li>\u00a0The point where the horizontal line [latex]y=0[\/latex] (the [latex]x[\/latex]-axis) meets the graph is at the point (3, 0). Therefore, [latex]x=3[\/latex] when [latex]f(x)=0[\/latex].<\/li>\n<li>\u00a0The point where the horizontal line [latex]y=6[\/latex] meets the graph is at the point (0, 6). Therefore, [latex]x=0[\/latex] when [latex]f(x)=6[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Use the graph of the function [latex]y=g(x)[\/latex] to determine the corresponding [latex]x[\/latex]-values when:<\/p>\n<ol>\n<li>\u00a0[latex]g(x)=4[\/latex]<\/li>\n<li>\u00a0[latex]g(x)=0[\/latex]<\/li>\n<li>\u00a0[latex]g(x)=-4[\/latex]<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1048\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/10003434\/gxx%5E3-2x-293x300.png\" alt=\"Graph of a cubic function where the line y=4 intersects at x=2, y=0 intersects at x= -1.5, 0, and 1.5, y=-4 intersects at x=-2\" width=\"368\" height=\"377\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm635\">Show Answer<\/span><\/p>\n<div id=\"qhjm635\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a0[latex]x=2[\/latex] when [latex]g(x)=4[\/latex]<\/li>\n<li>\u00a0[latex]x=-1.5, 0, 1.5[\/latex] when [latex]g(x)=0[\/latex]<\/li>\n<li>\u00a0[latex]x=-2[\/latex] when [latex]g(x)=-4[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>For [latex]T(x)=-2x+7[\/latex],\u00a0determine the corresponding [latex]x[\/latex]-values when:<\/p>\n<ol>\n<li>\u00a0[latex]T(x)=1[\/latex]<\/li>\n<li>\u00a0[latex]T(x)=-1[\/latex]<\/li>\n<li>\u00a0[latex]T(x)=9[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>Set\u00a0[latex]T(x)[\/latex] equal to the given values and solve the resulting equation for [latex]x[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">T(x)=1<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">T(x)=-1<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">T(x)=9<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">1.<\/p>\n<p>[latex]T(x)=-2x+7,\\;\\;1=-2x+7,\\;\\;-6=-2x,\\;\\;3=x[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">2.<\/p>\n<p>[latex]T(x)=-2x+7,\\;\\;-1=-2x+7,\\;\\;-8=-2x,\\;\\;4=x[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">\u00a03.<\/p>\n<p>[latex]T(x)=-2x+7,\\;\\;9=-2x+7,\\;\\;2=-2x,\\;\\;-1=x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>For [latex]T(x)=3x-2[\/latex],\u00a0determine the corresponding [latex]x[\/latex]-values when:<\/p>\n<ol>\n<li>\u00a0[latex]T(x)=7[\/latex]<\/li>\n<li>\u00a0[latex]T(x)=0[\/latex]<\/li>\n<li>\u00a0[latex]T(x)=-11[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm757\">Show Answer<\/span><\/p>\n<div id=\"qhjm757\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a0[latex]x=3[\/latex]<\/li>\n<li>\u00a0[latex]x=\\frac{2}{3}[\/latex]<\/li>\n<li>\u00a0[latex]x=-3[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/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-863\">\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>The Meaning of the Algebraic Form of a Function. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/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":422608,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"The Meaning of the Algebraic Form of a Function\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-863","chapter","type-chapter","status-publish","hentry"],"part":613,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/863","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":30,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/863\/revisions"}],"predecessor-version":[{"id":4896,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/863\/revisions\/4896"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/613"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/863\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=863"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=863"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=863"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=863"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}