{"id":1911,"date":"2016-11-02T22:15:27","date_gmt":"2016-11-02T22:15:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1911"},"modified":"2025-10-13T20:44:08","modified_gmt":"2025-10-13T20:44:08","slug":"summary-rational-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/chapter\/summary-rational-functions\/","title":{"raw":"Summary: Rational Functions","rendered":"Summary: Rational Functions"},"content":{"raw":"<section id=\"fs-id1165137659195\" class=\"key-equations\">\r\n<h1>Key Equations<\/h1>\r\n<table id=\"eip-id1362369\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>Rational Function<\/td>\r\n<td>[latex]f\\left(x\\right)=\\dfrac{P\\left(x\\right)}{Q\\left(x\\right)}=\\dfrac{{a}_{p}{x}^{p}+{a}_{p - 1}{x}^{p - 1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q - 1}{x}^{q - 1}+...+{b}_{1}x+{b}_{0}}, Q\\left(x\\right)\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137793507\" class=\"key-concepts\">\r\n<h1>Key Concepts<\/h1>\r\n<ul id=\"fs-id1165137603314\">\r\n \t<li>We can use arrow notation to describe local behavior and end behavior of the toolkit functions [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] and [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex].<\/li>\r\n \t<li>A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote.<\/li>\r\n \t<li>Application problems involving rates and concentrations often involve rational functions.<\/li>\r\n \t<li>The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.<\/li>\r\n \t<li>The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.<\/li>\r\n \t<li>A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero.<\/li>\r\n \t<li>A rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.<\/li>\r\n \t<li>Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior.<\/li>\r\n \t<li>If a rational function has <em>x<\/em>-intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex], vertical asymptotes at [latex]x={v}_{1},{v}_{2},\\dots ,{v}_{m}[\/latex], and no [latex]{x}_{i}=\\text{any }{v}_{j}[\/latex], then the function can be written in the form\u00a0[latex]f\\left(x\\right)=a\\dfrac{{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}}{{\\left(x-{v}_{1}\\right)}^{{q}_{1}}{\\left(x-{v}_{2}\\right)}^{{q}_{2}}\\cdots {\\left(x-{v}_{m}\\right)}^{{q}_{n}}}[\/latex]<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137758530\" class=\"definition\">\r\n \t<dt><strong>arrow notation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135154402\">a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135154407\" class=\"definition\">\r\n \t<dt><strong>horizontal asymptote<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135154413\">a horizontal line [latex]y=b[\/latex]\u00a0where the graph approaches the line as the inputs increase or decrease without bound.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135192626\" class=\"definition\">\r\n \t<dt><strong>rational function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134401081\">a function that can be written as the ratio of two polynomials<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134401085\" class=\"definition\">\r\n \t<dt><strong>removable discontinuity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134401090\">a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137426312\" class=\"definition\">\r\n \t<dt><strong>vertical asymptote<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137426317\">a vertical line [latex]x=a[\/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]a[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>","rendered":"<section id=\"fs-id1165137659195\" class=\"key-equations\">\n<h1>Key Equations<\/h1>\n<table id=\"eip-id1362369\" summary=\"..\">\n<tbody>\n<tr>\n<td>Rational Function<\/td>\n<td>[latex]f\\left(x\\right)=\\dfrac{P\\left(x\\right)}{Q\\left(x\\right)}=\\dfrac{{a}_{p}{x}^{p}+{a}_{p - 1}{x}^{p - 1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q - 1}{x}^{q - 1}+...+{b}_{1}x+{b}_{0}}, Q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137793507\" class=\"key-concepts\">\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1165137603314\">\n<li>We can use arrow notation to describe local behavior and end behavior of the toolkit functions [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] and [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex].<\/li>\n<li>A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote.<\/li>\n<li>Application problems involving rates and concentrations often involve rational functions.<\/li>\n<li>The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.<\/li>\n<li>The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.<\/li>\n<li>A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero.<\/li>\n<li>A rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.<\/li>\n<li>Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior.<\/li>\n<li>If a rational function has <em>x<\/em>-intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex], vertical asymptotes at [latex]x={v}_{1},{v}_{2},\\dots ,{v}_{m}[\/latex], and no [latex]{x}_{i}=\\text{any }{v}_{j}[\/latex], then the function can be written in the form\u00a0[latex]f\\left(x\\right)=a\\dfrac{{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}}{{\\left(x-{v}_{1}\\right)}^{{q}_{1}}{\\left(x-{v}_{2}\\right)}^{{q}_{2}}\\cdots {\\left(x-{v}_{m}\\right)}^{{q}_{n}}}[\/latex]<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137758530\" class=\"definition\">\n<dt><strong>arrow notation<\/strong><\/dt>\n<dd id=\"fs-id1165135154402\">a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135154407\" class=\"definition\">\n<dt><strong>horizontal asymptote<\/strong><\/dt>\n<dd id=\"fs-id1165135154413\">a horizontal line [latex]y=b[\/latex]\u00a0where the graph approaches the line as the inputs increase or decrease without bound.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135192626\" class=\"definition\">\n<dt><strong>rational function<\/strong><\/dt>\n<dd id=\"fs-id1165134401081\">a function that can be written as the ratio of two polynomials<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134401085\" class=\"definition\">\n<dt><strong>removable discontinuity<\/strong><\/dt>\n<dd id=\"fs-id1165134401090\">a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137426312\" class=\"definition\">\n<dt><strong>vertical asymptote<\/strong><\/dt>\n<dd id=\"fs-id1165137426317\">a vertical line [latex]x=a[\/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]a[\/latex]<\/dd>\n<\/dl>\n<\/div>\n<\/section>\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-1911\">\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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/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":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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