{"id":2583,"date":"2022-06-07T23:56:40","date_gmt":"2022-06-07T23:56:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2583"},"modified":"2026-01-18T01:29:10","modified_gmt":"2026-01-18T01:29:10","slug":"7-1-rational-functions-and-their-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/7-1-rational-functions-and-their-graphs\/","title":{"raw":"7.1.1: Rational Functions and Their Graphs","rendered":"7.1.1: Rational Functions and Their Graphs"},"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 definition of a rational function<\/li>\r\n \t<li>Use Desmos to graph rational functions<\/li>\r\n \t<li>Determine the Domain and Range of rational functions<\/li>\r\n \t<li>Determine the vertical and horizontal asymptotes of rational functions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Definition of Rational Functions<\/h2>\r\nA <em><strong>rational function<\/strong><\/em> has the form [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex], where [latex]P(x)[\/latex] and\u00a0[latex]Q(x)[\/latex] are polynomials. In addition, [latex]Q(x)\\neq 0[\/latex]. The expression [latex]\\dfrac{P(x)}{Q(x)}[\/latex] is called a <em><strong>rational expression<\/strong><\/em>. Examples of rational functions include:\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]f(x)=\\dfrac{1}{x}[\/latex],\u00a0 [latex]f(x)=\\dfrac{5x-3}{x^2-1}[\/latex], [latex]h(x)=\\dfrac{x^2-5}{x^3+2x^2+7}[\/latex], [latex]g(x)=\\dfrac{7x^3-5x}{x^2-5}[\/latex].<\/span>\r\n<h2>Graphs of Rational Functions<\/h2>\r\nWe can use <a href=\"http:\/\/desmos.com\/calculator\">Desmos<\/a> to graph rational functions. Try it out for yourself. Figure 1 shows some examples.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\" colspan=\"2\">Examples of Rational Functions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><img class=\"aligncenter wp-image-3302 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and upward, the other right and downward, each approaching the same vertical and horizontal lines without intersecting them.\" width=\"300\" height=\"300\" \/><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><img class=\"aligncenter wp-image-3303 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/15152944\/desmos-graph-2022-07-15T092927.854-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and upward, the other right and upward. A third curve branch opens downward. Each curve approaches the same vertical and horizontal lines without intersecting them.\" width=\"300\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><img class=\"aligncenter wp-image-3304 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and downward, the other right and downward. A third curve branch opens upward. Each curve approaches the same vertical and horizontal lines.\" width=\"300\" height=\"300\" \/><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><img class=\"aligncenter wp-image-3307 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening southwest and downward with one side approaching a diagonal line and the other side approaching a vertical line, the other northeast and downward with one side approaching the same diagonal line and the other side approaching another vertical line. A third curve branch opens upward between the two curve branches and approaching the two vertical lines.\" width=\"300\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: left;\" colspan=\"2\">Figure 1. Examples of the graphs of rational functions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Domain and Range<\/h2>\r\nEach function has a domain and a range. The domain is the set of all [latex]x[\/latex]-values, while the range is the set of all\u00a0[latex]y[\/latex]-values.\r\n\r\nThe domain for the function [latex]f(x)=\\dfrac{x-1}{x+4}[\/latex] is all real numbers except [latex]x=-4[\/latex]. It can be written in interval notation as [latex]x\\in(-\\infty,\\,-4)\\cup(-4,\\,+\\infty)[\/latex] or in set notation as [latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq -4\\}[\/latex]. The point [latex]x=-4[\/latex] is excluded from the domain because at that point the function value is undefined: [latex]f(-4)=\\dfrac{-4-1}{-4+4}=\\dfrac{-5}{0}[\/latex], and we can't divide by zero. [latex]x=-4[\/latex] is called a <em><strong>restriction on the domain<\/strong><\/em>. The line [latex]x=-4[\/latex] is a vertical asymptote (Figure 1: top left).\r\n\r\nThe range of\u00a0the function [latex]f(x)=\\dfrac{x-1}{x+4}[\/latex] is all real numbers except [latex]y=1[\/latex]. It can be written in interval notation as [latex]y\\in(-\\infty,\\,1)\\cup(1,\\,+\\infty)[\/latex] or in set notation as [latex]\\{y\\;|\\;y \\in \\mathbb{R}, y\\neq 1\\}[\/latex]. There is a horizontal asymptote at [latex]y=1[\/latex]\u00a0(Figure 1: top left). Horizontal asymptotes appear on the graphs of rational functions [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] when the degree of\u00a0[latex]P(x)[\/latex] is less than or equal to the degree of\u00a0[latex]Q(x)[\/latex]. In this case the degrees are equal at 1. When the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients of\u00a0[latex]P(x)[\/latex] and\u00a0[latex]Q(x)[\/latex]. In this case, the horizontal asymptote is [latex]y=\\dfrac{1}{1}=1[\/latex].\r\n\r\nThe domain for the function [latex]f(x)=\\dfrac{1}{x^2-1}[\/latex] is all real numbers except [latex]x=-1[\/latex] and [latex]x=1[\/latex]. It can be written in interval notation as [latex]x\\in(-\\infty,\\,-1)\\cup(-1,\\,1)\\cup(1,\\,+\\infty)[\/latex] or in set notation as [latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq \\pm1\\}[\/latex]. The points [latex]x=-1[\/latex] and [latex]x=1[\/latex] are excluded from the domain because at those points the function value is undefined. [latex]x=-1[\/latex] and [latex]x=1[\/latex] are\u00a0<em><strong>restrictions on the domain<\/strong><\/em>. The lines [latex]x=-1[\/latex] and [latex]x=1[\/latex] are vertical asymptotes (Figure 1: top right).\r\n\r\nThe range for the function [latex]f(x)=\\dfrac{1}{x^2-1}[\/latex] can be seen from the graph to exclude the [latex]y[\/latex]-values greater than \u20131 and less than or equal to 0. That means the domain is all real numbers except [latex]-1&lt;y\u22640[\/latex]. It can be written in interval notation as [latex]y\\in(-\\infty,\\,-1]\\cup(0,\\,+\\infty)[\/latex] or in set notation as [latex]\\{y\\;|\\;y \\in \\mathbb{R}[\/latex], \\ [latex]-1&lt;y\u22640\\}[\/latex]. The symbol \\ means except. There is a horizontal asymptote at [latex]y=0[\/latex] (Figure 1: top right). There will always be a horizontal asymptote at [latex]y=0[\/latex] when the degree of [latex]P(x)[\/latex] is less than the degree of\u00a0[latex]Q(x)[\/latex]. In this case,\u00a0the degree of [latex]P(x)[\/latex] is 0, which is less than the degree of\u00a0[latex]Q(x)[\/latex], which is 2.\r\n<div class=\"textbox shaded\">\r\n<h2>RESTRICTIONS ON THE DOMAIN<\/h2>\r\nThe rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] has restrictions on the domain when\u00a0[latex]Q(x)=0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nState the domain and range of the function [latex]f(x)=\\dfrac{x^2+3x-11}{x^2-4}[\/latex].\r\n\r\n[caption id=\"attachment_4431\" align=\"alignright\" width=\"262\"]<img class=\"wp-image-4431\" style=\"font-size: 16px; orphans: 1; background-color: #ffffff;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and downward, the other right and downward. A third curve branch opens upward. Each curve approaches the same vertical and horizontal lines without intersecting them, leaving a rectangular region with no curves.\" width=\"262\" height=\"262\" \/> Zoomed in view of [latex]f(x)=\\dfrac{x^{2}+3x-11}{x^2-4}[\/latex][\/caption]\r\n<h4>Solution<\/h4>\r\nThis function is graphed in figure 1 (bottom left). It has vertical asymptotes at [latex]x=\\pm2[\/latex], therefore\u00a0[latex]x=\\pm2[\/latex] are restrictions on the domain.\r\n\r\nIn interval notation Domain = [latex](-\\infty,\\,-2)\\cup(-2,\\,2)\\cup(2,\\,+\\infty)[\/latex]\r\n\r\nIn set notation Domain =\u00a0[latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq \\pm2\\}[\/latex]\r\n\r\n&nbsp;\r\n\r\nThe graph has a horizontal asymptote at [latex]y=1[\/latex] but this asymptote is crossed by the function just at [latex]x=2.333[\/latex] so must be included in the range. There is however a gap in the range values from the highest point on the lower right part of the graph and the turning point on the upper part of the curve. We can use Desmos to determine the approximate value of these points by moving the curser over the graph. So, the range is [latex]y\\in(-\\infty,\\,1.424]\\cup[2.326,\\,+\\infty)[\/latex] in interval notation, and [latex]\\{y\\;|\\;y \\in \\mathbb{R}, y \\not \\in [1.424, 2.326]\\}[\/latex] in set notation.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\n<ol>\r\n \t<li>Use Desmos to graph the function [latex]f(x)=\\dfrac{3x+4}{x-2}[\/latex].<\/li>\r\n \t<li>What is the vertical asymptote?<\/li>\r\n \t<li>What is the horizontal asymptote?<\/li>\r\n \t<li>Determine the domain of the function. Write it in set notation.<\/li>\r\n \t<li>Determine the range of the function. Write it in interval notation.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm858\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm858\"]\r\n<ol>\r\n \t<li><img class=\"aligncenter wp-image-3321 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and downward, the other right and upward. Each curve approaches the same vertical and horizontal asymptotes without intersecting them.\" width=\"300\" height=\"300\" \/><\/li>\r\n \t<li>[latex]x=2[\/latex]<\/li>\r\n \t<li>[latex]y=3[\/latex]<\/li>\r\n \t<li>Domain =\u00a0[latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq \\pm2\\}[\/latex]<\/li>\r\n \t<li>Range =\u00a0[latex]y\\in(-\\infty,\\,3)\\cup(3,\\,+\\infty)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Vertical Asymptotes<\/h2>\r\nHave you noticed the relationship between the vertical asymptotes and the restrictions on the domain?\r\n\r\nFor example,\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%;\"><strong>Function<\/strong><\/th>\r\n<th style=\"width: 33.3333%;\"><strong>Restriction on the domain<\/strong><\/th>\r\n<th style=\"width: 33.3333%;\"><strong>Vertical asymptote<\/strong><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[latex]f(x)=\\dfrac{x-1}{x+4}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]x=-4[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]x=-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[latex]f(x)=\\dfrac{1}{x^2-1}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]x=\\pm1[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]x=\\pm1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[latex]f(x)=\\dfrac{x^2+3x-11}{x^2-4}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]x=\\pm2[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">\u00a0[latex]x=\\pm2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nA vertical asymptote is a vertical line that passes through the restricted value of the domain. The vertical asymptote can never be crossed by the graph, since the graph is undefined at that value.\r\n\r\nWe can easily find any vertical asymptotes by determining the restrictions on the domain. For a rational function [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] the domain will be restricted when the denominator, [latex]Q(x)[\/latex], is equal to zero. This is because division by zero is undefined.\r\n<div class=\"textbox shaded\">\r\n<h3>VERTICAL ASYMPTOTES<\/h3>\r\nThe rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] will have vertical asymptotes at\u00a0[latex]Q(x)=0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nDetermine the vertical asymptote(s) of the function [latex]f(x)=\\dfrac{3x}{x^2-3x-4}[\/latex].\r\n<h4>Solution<\/h4>\r\nVertical asymptotes occur at restrictions on the domain. The domain is restricted at [latex]x[\/latex]-values that cause the denominator of the function to equal zero.\r\n\r\nSo we set the denominator to zero and solve for [latex]x[\/latex]:\r\n<p style=\"text-align: center;\">[latex]x^2-3x-4=0[\/latex]<\/p>\r\nTo solve for [latex]x[\/latex], we can factor the quadratic then use the zero product property:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x^2-3x-4&amp;=0\\\\\\\\(x-4)(x+1)&amp;=0\\\\\\\\x=4,\\,x&amp;=-1\\end{aligned}[\/latex]<\/p>\r\nThe vertical asymptotes are [latex]x=4[\/latex] and [latex]x=-1[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDetermine the vertical asymptote of the function [latex]f(x)=\\dfrac{2x-5}{x^2-9}[\/latex].\r\n\r\n[reveal-answer q=\"hjm607\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm607\"][latex]x=\\pm3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Horizontal Asymptotes<\/h2>\r\nHorizontal asymptotes appear on the graph of a rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] when the degree of\u00a0[latex]P(x)[\/latex] is less than or equal to the degree of\u00a0[latex]Q(x)[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>HORIZONTAL ASYMPTOTES<\/h3>\r\nThe rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] will have a horizontal asymptote when\u00a0the degree of\u00a0[latex]P(x)[\/latex] is less than or equal to the degree of\u00a0[latex]Q(x)[\/latex].\r\n\r\nIf the degree [latex]P(x)[\/latex] is less than the degree of\u00a0[latex]Q(x)[\/latex], the horizontal asymptote will be [latex]y=0[\/latex].\r\n\r\nIf the degree [latex]P(x)[\/latex] is equal to the degree of\u00a0[latex]Q(x)[\/latex], the horizontal asymptote will be [latex]y=\\dfrac{\\text{Leading coefficient of }P(x)}{\\text{Leading coefficient of }Q(x)}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nDetermine the horizontal asymptotes of the graphs of the function:\r\n<ol>\r\n \t<li>[latex]f(x)=\\dfrac{4x-5}{2x+7}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{9x^2-1}{3x^2+2}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{x-3}{2x^2+6x-3}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{x^3-5}{x+7}[\/latex]<\/li>\r\n<\/ol>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<ol>\r\n \t<li>[latex]P(x)=4x-5[\/latex] has degree 1. [latex]Q(x)=2x+7[\/latex] has degree 1. Since the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients: [latex]y=\\dfrac{4}{2}=2[\/latex].<\/li>\r\n \t<li>[latex]P(x)=9x-1[\/latex] has degree 2. [latex]Q(x)=3x^2+2[\/latex] has degree 2. Since the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients: [latex]y=\\dfrac{9}{3}=3[\/latex].<\/li>\r\n \t<li>[latex]P(x)=x-3[\/latex] has degree 1. [latex]Q(x)=2x^2+6x-3[\/latex] has degree 2. Since the degree of [latex]P(x)[\/latex] is less than the degree of [latex]Q(x)[\/latex], the horizontal asymptote is [latex]y=0[\/latex].<\/li>\r\n \t<li>[latex]P(x)=x^3-5[\/latex] has degree 3. [latex]Q(x)=x+7[\/latex] has degree 1. Since the degree of [latex]P(x)[\/latex] is greater than the degree of [latex]Q(x)[\/latex], there is no horizontal asymptote.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nDetermine the horizontal asymptotes of the graphs of the function:\r\n<ol>\r\n \t<li>[latex]f(x)=\\dfrac{14x-3}{7x+4}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{2x^2-1}{3x^3+2}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{x^3-3}{2x^2}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{x^2-5}{4x^2+7}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm098\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm098\"]\r\n<ol>\r\n \t<li>[latex]y=2[\/latex]<\/li>\r\n \t<li>[latex]y=0[\/latex]<\/li>\r\n \t<li>There is no horizontal asymptote<\/li>\r\n \t<li>[latex]y=\\dfrac{1}{4}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\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 definition of a rational function<\/li>\n<li>Use Desmos to graph rational functions<\/li>\n<li>Determine the Domain and Range of rational functions<\/li>\n<li>Determine the vertical and horizontal asymptotes of rational functions<\/li>\n<\/ul>\n<\/div>\n<h2>Definition of Rational Functions<\/h2>\n<p>A <em><strong>rational function<\/strong><\/em> has the form [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex], where [latex]P(x)[\/latex] and\u00a0[latex]Q(x)[\/latex] are polynomials. In addition, [latex]Q(x)\\neq 0[\/latex]. The expression [latex]\\dfrac{P(x)}{Q(x)}[\/latex] is called a <em><strong>rational expression<\/strong><\/em>. Examples of rational functions include:\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]f(x)=\\dfrac{1}{x}[\/latex],\u00a0 [latex]f(x)=\\dfrac{5x-3}{x^2-1}[\/latex], [latex]h(x)=\\dfrac{x^2-5}{x^3+2x^2+7}[\/latex], [latex]g(x)=\\dfrac{7x^3-5x}{x^2-5}[\/latex].<\/span><\/p>\n<h2>Graphs of Rational Functions<\/h2>\n<p>We can use <a href=\"http:\/\/desmos.com\/calculator\">Desmos<\/a> to graph rational functions. Try it out for yourself. Figure 1 shows some examples.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\" colspan=\"2\">Examples of Rational Functions<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3302 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and upward, the other right and downward, each approaching the same vertical and horizontal lines without intersecting them.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T092612.998.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 50%; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3303 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/15152944\/desmos-graph-2022-07-15T092927.854-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and upward, the other right and upward. A third curve branch opens downward. Each curve approaches the same vertical and horizontal lines without intersecting them.\" width=\"300\" height=\"300\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3304 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and downward, the other right and downward. A third curve branch opens upward. Each curve approaches the same vertical and horizontal lines.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T093326.964.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 50%; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3307 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening southwest and downward with one side approaching a diagonal line and the other side approaching a vertical line, the other northeast and downward with one side approaching the same diagonal line and the other side approaching another vertical line. A third curve branch opens upward between the two curve branches and approaching the two vertical lines.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T105745.272.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: left;\" colspan=\"2\">Figure 1. Examples of the graphs of rational functions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Domain and Range<\/h2>\n<p>Each function has a domain and a range. The domain is the set of all [latex]x[\/latex]-values, while the range is the set of all\u00a0[latex]y[\/latex]-values.<\/p>\n<p>The domain for the function [latex]f(x)=\\dfrac{x-1}{x+4}[\/latex] is all real numbers except [latex]x=-4[\/latex]. It can be written in interval notation as [latex]x\\in(-\\infty,\\,-4)\\cup(-4,\\,+\\infty)[\/latex] or in set notation as [latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq -4\\}[\/latex]. The point [latex]x=-4[\/latex] is excluded from the domain because at that point the function value is undefined: [latex]f(-4)=\\dfrac{-4-1}{-4+4}=\\dfrac{-5}{0}[\/latex], and we can&#8217;t divide by zero. [latex]x=-4[\/latex] is called a <em><strong>restriction on the domain<\/strong><\/em>. The line [latex]x=-4[\/latex] is a vertical asymptote (Figure 1: top left).<\/p>\n<p>The range of\u00a0the function [latex]f(x)=\\dfrac{x-1}{x+4}[\/latex] is all real numbers except [latex]y=1[\/latex]. It can be written in interval notation as [latex]y\\in(-\\infty,\\,1)\\cup(1,\\,+\\infty)[\/latex] or in set notation as [latex]\\{y\\;|\\;y \\in \\mathbb{R}, y\\neq 1\\}[\/latex]. There is a horizontal asymptote at [latex]y=1[\/latex]\u00a0(Figure 1: top left). Horizontal asymptotes appear on the graphs of rational functions [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] when the degree of\u00a0[latex]P(x)[\/latex] is less than or equal to the degree of\u00a0[latex]Q(x)[\/latex]. In this case the degrees are equal at 1. When the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients of\u00a0[latex]P(x)[\/latex] and\u00a0[latex]Q(x)[\/latex]. In this case, the horizontal asymptote is [latex]y=\\dfrac{1}{1}=1[\/latex].<\/p>\n<p>The domain for the function [latex]f(x)=\\dfrac{1}{x^2-1}[\/latex] is all real numbers except [latex]x=-1[\/latex] and [latex]x=1[\/latex]. It can be written in interval notation as [latex]x\\in(-\\infty,\\,-1)\\cup(-1,\\,1)\\cup(1,\\,+\\infty)[\/latex] or in set notation as [latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq \\pm1\\}[\/latex]. The points [latex]x=-1[\/latex] and [latex]x=1[\/latex] are excluded from the domain because at those points the function value is undefined. [latex]x=-1[\/latex] and [latex]x=1[\/latex] are\u00a0<em><strong>restrictions on the domain<\/strong><\/em>. The lines [latex]x=-1[\/latex] and [latex]x=1[\/latex] are vertical asymptotes (Figure 1: top right).<\/p>\n<p>The range for the function [latex]f(x)=\\dfrac{1}{x^2-1}[\/latex] can be seen from the graph to exclude the [latex]y[\/latex]-values greater than \u20131 and less than or equal to 0. That means the domain is all real numbers except [latex]-1<y\u22640[\/latex]. It can be written in interval notation as [latex]y\\in(-\\infty,\\,-1]\\cup(0,\\,+\\infty)[\/latex] or in set notation as [latex]\\{y\\;|\\;y \\in \\mathbb{R}[\/latex], \\ [latex]-1<y\u22640\\}[\/latex]. The symbol \\ means except. There is a horizontal asymptote at [latex]y=0[\/latex] (Figure 1: top right). There will always be a horizontal asymptote at [latex]y=0[\/latex] when the degree of [latex]P(x)[\/latex] is less than the degree of\u00a0[latex]Q(x)[\/latex]. In this case,\u00a0the degree of [latex]P(x)[\/latex] is 0, which is less than the degree of\u00a0[latex]Q(x)[\/latex], which is 2.\n\n\n<div class=\"textbox shaded\">\n<h2>RESTRICTIONS ON THE DOMAIN<\/h2>\n<p>The rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] has restrictions on the domain when\u00a0[latex]Q(x)=0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>State the domain and range of the function [latex]f(x)=\\dfrac{x^2+3x-11}{x^2-4}[\/latex].<\/p>\n<div id=\"attachment_4431\" style=\"width: 272px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4431\" class=\"wp-image-4431\" style=\"font-size: 16px; orphans: 1; background-color: #ffffff;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and downward, the other right and downward. A third curve branch opens upward. Each curve approaches the same vertical and horizontal lines without intersecting them, leaving a rectangular region with no curves.\" width=\"262\" height=\"262\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2023-11-17T125043.108.png 800w\" sizes=\"auto, (max-width: 262px) 100vw, 262px\" \/><\/p>\n<p id=\"caption-attachment-4431\" class=\"wp-caption-text\">Zoomed in view of [latex]f(x)=\\dfrac{x^{2}+3x-11}{x^2-4}[\/latex]<\/p>\n<\/div>\n<h4>Solution<\/h4>\n<p>This function is graphed in figure 1 (bottom left). It has vertical asymptotes at [latex]x=\\pm2[\/latex], therefore\u00a0[latex]x=\\pm2[\/latex] are restrictions on the domain.<\/p>\n<p>In interval notation Domain = [latex](-\\infty,\\,-2)\\cup(-2,\\,2)\\cup(2,\\,+\\infty)[\/latex]<\/p>\n<p>In set notation Domain =\u00a0[latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq \\pm2\\}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The graph has a horizontal asymptote at [latex]y=1[\/latex] but this asymptote is crossed by the function just at [latex]x=2.333[\/latex] so must be included in the range. There is however a gap in the range values from the highest point on the lower right part of the graph and the turning point on the upper part of the curve. We can use Desmos to determine the approximate value of these points by moving the curser over the graph. So, the range is [latex]y\\in(-\\infty,\\,1.424]\\cup[2.326,\\,+\\infty)[\/latex] in interval notation, and [latex]\\{y\\;|\\;y \\in \\mathbb{R}, y \\not \\in [1.424, 2.326]\\}[\/latex] in set notation.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<ol>\n<li>Use Desmos to graph the function [latex]f(x)=\\dfrac{3x+4}{x-2}[\/latex].<\/li>\n<li>What is the vertical asymptote?<\/li>\n<li>What is the horizontal asymptote?<\/li>\n<li>Determine the domain of the function. Write it in set notation.<\/li>\n<li>Determine the range of the function. Write it in interval notation.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm858\">Show Answer<\/span><\/p>\n<div id=\"qhjm858\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3321 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-300x300.png\" alt=\"Two separate curve branches in opposite quadrants: one opening left and downward, the other right and upward. Each curve approaches the same vertical and horizontal asymptotes without intersecting them.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-15T125849.437.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/li>\n<li>[latex]x=2[\/latex]<\/li>\n<li>[latex]y=3[\/latex]<\/li>\n<li>Domain =\u00a0[latex]\\{x\\;|\\;x \\in \\mathbb{R}, x\\neq \\pm2\\}[\/latex]<\/li>\n<li>Range =\u00a0[latex]y\\in(-\\infty,\\,3)\\cup(3,\\,+\\infty)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Vertical Asymptotes<\/h2>\n<p>Have you noticed the relationship between the vertical asymptotes and the restrictions on the domain?<\/p>\n<p>For example,<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%;\"><strong>Function<\/strong><\/th>\n<th style=\"width: 33.3333%;\"><strong>Restriction on the domain<\/strong><\/th>\n<th style=\"width: 33.3333%;\"><strong>Vertical asymptote<\/strong><\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">[latex]f(x)=\\dfrac{x-1}{x+4}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]x=-4[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]x=-4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">[latex]f(x)=\\dfrac{1}{x^2-1}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]x=\\pm1[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]x=\\pm1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">[latex]f(x)=\\dfrac{x^2+3x-11}{x^2-4}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]x=\\pm2[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">\u00a0[latex]x=\\pm2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A vertical asymptote is a vertical line that passes through the restricted value of the domain. The vertical asymptote can never be crossed by the graph, since the graph is undefined at that value.<\/p>\n<p>We can easily find any vertical asymptotes by determining the restrictions on the domain. For a rational function [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] the domain will be restricted when the denominator, [latex]Q(x)[\/latex], is equal to zero. This is because division by zero is undefined.<\/p>\n<div class=\"textbox shaded\">\n<h3>VERTICAL ASYMPTOTES<\/h3>\n<p>The rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] will have vertical asymptotes at\u00a0[latex]Q(x)=0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Determine the vertical asymptote(s) of the function [latex]f(x)=\\dfrac{3x}{x^2-3x-4}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Vertical asymptotes occur at restrictions on the domain. The domain is restricted at [latex]x[\/latex]-values that cause the denominator of the function to equal zero.<\/p>\n<p>So we set the denominator to zero and solve for [latex]x[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]x^2-3x-4=0[\/latex]<\/p>\n<p>To solve for [latex]x[\/latex], we can factor the quadratic then use the zero product property:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x^2-3x-4&=0\\\\\\\\(x-4)(x+1)&=0\\\\\\\\x=4,\\,x&=-1\\end{aligned}[\/latex]<\/p>\n<p>The vertical asymptotes are [latex]x=4[\/latex] and [latex]x=-1[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Determine the vertical asymptote of the function [latex]f(x)=\\dfrac{2x-5}{x^2-9}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm607\">Show Answer<\/span><\/p>\n<div id=\"qhjm607\" class=\"hidden-answer\" style=\"display: none\">[latex]x=\\pm3[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Horizontal Asymptotes<\/h2>\n<p>Horizontal asymptotes appear on the graph of a rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] when the degree of\u00a0[latex]P(x)[\/latex] is less than or equal to the degree of\u00a0[latex]Q(x)[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>HORIZONTAL ASYMPTOTES<\/h3>\n<p>The rational function\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] will have a horizontal asymptote when\u00a0the degree of\u00a0[latex]P(x)[\/latex] is less than or equal to the degree of\u00a0[latex]Q(x)[\/latex].<\/p>\n<p>If the degree [latex]P(x)[\/latex] is less than the degree of\u00a0[latex]Q(x)[\/latex], the horizontal asymptote will be [latex]y=0[\/latex].<\/p>\n<p>If the degree [latex]P(x)[\/latex] is equal to the degree of\u00a0[latex]Q(x)[\/latex], the horizontal asymptote will be [latex]y=\\dfrac{\\text{Leading coefficient of }P(x)}{\\text{Leading coefficient of }Q(x)}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Determine the horizontal asymptotes of the graphs of the function:<\/p>\n<ol>\n<li>[latex]f(x)=\\dfrac{4x-5}{2x+7}[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{9x^2-1}{3x^2+2}[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{x-3}{2x^2+6x-3}[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{x^3-5}{x+7}[\/latex]<\/li>\n<\/ol>\n<h4><strong>Solution<\/strong><\/h4>\n<ol>\n<li>[latex]P(x)=4x-5[\/latex] has degree 1. [latex]Q(x)=2x+7[\/latex] has degree 1. Since the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients: [latex]y=\\dfrac{4}{2}=2[\/latex].<\/li>\n<li>[latex]P(x)=9x-1[\/latex] has degree 2. [latex]Q(x)=3x^2+2[\/latex] has degree 2. Since the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients: [latex]y=\\dfrac{9}{3}=3[\/latex].<\/li>\n<li>[latex]P(x)=x-3[\/latex] has degree 1. [latex]Q(x)=2x^2+6x-3[\/latex] has degree 2. Since the degree of [latex]P(x)[\/latex] is less than the degree of [latex]Q(x)[\/latex], the horizontal asymptote is [latex]y=0[\/latex].<\/li>\n<li>[latex]P(x)=x^3-5[\/latex] has degree 3. [latex]Q(x)=x+7[\/latex] has degree 1. Since the degree of [latex]P(x)[\/latex] is greater than the degree of [latex]Q(x)[\/latex], there is no horizontal asymptote.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Determine the horizontal asymptotes of the graphs of the function:<\/p>\n<ol>\n<li>[latex]f(x)=\\dfrac{14x-3}{7x+4}[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{2x^2-1}{3x^3+2}[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{x^3-3}{2x^2}[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{x^2-5}{4x^2+7}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm098\">Show Answer<\/span><\/p>\n<div id=\"qhjm098\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]y=2[\/latex]<\/li>\n<li>[latex]y=0[\/latex]<\/li>\n<li>There is no horizontal asymptote<\/li>\n<li>[latex]y=\\dfrac{1}{4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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-2583\">\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>Rational Functions and Their Graphs. <strong>Authored by<\/strong>: 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>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.desmos.com\/calculator\">http:\/\/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><li>All examples and Try Its: hjm858; hjm607; hjm098; . <strong>Authored by<\/strong>: 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><\/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":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Rational Functions and Their Graphs\",\"author\":\"Hazel McKenna \",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"www.desmos.com\/calculator\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All examples and Try Its: hjm858; hjm607; hjm098; \",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"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-2583","chapter","type-chapter","status-publish","hentry"],"part":2581,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2583","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":57,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2583\/revisions"}],"predecessor-version":[{"id":4828,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2583\/revisions\/4828"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2581"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2583\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2583"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2583"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2583"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2583"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}