{"id":84,"date":"2025-02-13T22:43:34","date_gmt":"2025-02-13T22:43:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions\/"},"modified":"2026-01-14T18:26:34","modified_gmt":"2026-01-14T18:26:34","slug":"rational-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions\/","title":{"raw":"Rational Functions: Learn It 1","rendered":"Rational Functions: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"text-align: justify;\">Utilize arrow notation to specify the behavior of rational functions.<\/li>\r\n \t<li style=\"text-align: justify;\">Find the domains of rational functions.<\/li>\r\n \t<li style=\"text-align: justify;\">Identify vertical and horizontal asymptotes.<\/li>\r\n \t<li style=\"text-align: justify;\">Identify slant asymptotes.<\/li>\r\n \t<li style=\"text-align: justify;\">Graph rational functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Rational Functions<\/h2>\r\nA <strong>rational function<\/strong> is a function that can be written as the quotient of two polynomial functions [latex]P\\left(x\\right) \\text{ and } Q\\left(x\\right)[\/latex].\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[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]<\/p>\r\nWhen a variable is present in the denominator of a rational expression, certain values of the variable may cause the denominator to equal zero. A rational expression with a zero in the denominator is not defined since we cannot divide by zero.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">We have seen the graphs of the basic <strong>reciprocal function<\/strong> and the squared reciprocal function from our study of toolkit functions.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213907\/CNX_Precalc_Figure_03_07_0012.jpg\" alt=\"Graphs of f(x)=1\/x and f(x)=1\/x^2\" width=\"731\" height=\"453\" \/>Several things are apparent if we examine the graph of [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex].\r\n<ol>\r\n \t<li>On the left branch of the graph, the curve approaches the [latex]x[\/latex]-axis [latex]\\left(y=0\\right) \\text{ as } x\\to -\\infty [\/latex].<\/li>\r\n \t<li>As the graph approaches [latex]x=0[\/latex] from the left, the curve drops, but as we approach zero from the right, the curve rises.<\/li>\r\n \t<li>Finally, on the right branch of the graph, the curves approaches the [latex]x[\/latex]<em>-<\/em>axis [latex]\\left(y=0\\right) \\text{ as } x\\to \\infty [\/latex].<\/li>\r\n<\/ol>\r\n<\/section>\r\n<h3>Arrow Notations<\/h3>\r\nWe use <strong>arrow notation<\/strong> to show that [latex]x[\/latex]\u00a0or [latex]f\\left(x\\right)[\/latex] is approaching a particular value.\r\n<table style=\"height: 240px; width: 100%;\">\r\n<thead>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"text-align: center; height: 15px; width: 99.0867%;\" colspan=\"2\">Arrow Notation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"text-align: center; height: 15px; width: 32.9933%;\">Symbol<\/th>\r\n<th style=\"text-align: center; height: 15px; width: 66.0934%;\">Meaning<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to {a}^{-}[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches [latex]a[\/latex] from the left ([latex]x \\lt a[\/latex] but close to [latex]a[\/latex])<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to {a}^{+}[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches [latex]a[\/latex] from the right ([latex]x \\lt a[\/latex] but close to [latex]a[\/latex])<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to \\infty[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches infinity ([latex]x[\/latex]\u00a0increases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to -\\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches negative infinity ([latex]x[\/latex]\u00a0decreases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]f\\left(x\\right)\\to \\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">the output approaches infinity (the output increases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]f\\left(x\\right)\\to -\\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">the output approaches negative infinity (the output decreases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">the output approaches [latex]a[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\" aria-label=\"Example\">Let\u2019s begin by looking at the reciprocal function, [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex].As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in the table below.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]\u20130.1[\/latex]<\/td>\r\n<td>[latex]\u20130.01[\/latex]<\/td>\r\n<td>[latex]\u20130.001[\/latex]<\/td>\r\n<td>[latex]\u20130.0001[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] <\/strong><\/td>\r\n<td>[latex]\u201310[\/latex]<\/td>\r\n<td>[latex]\u2013100[\/latex]<\/td>\r\n<td>[latex]\u20131000[\/latex]<\/td>\r\n<td>[latex]\u201310,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe write in arrow notation:\r\n<p style=\"text-align: center;\">[latex]\\text{as }x\\to {0}^{-},f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\r\n<p style=\"text-align: center;\">As [latex]x[\/latex] approaches [latex]0[\/latex] from the left (negative) side, [latex]f(x)[\/latex] will approach negative infinity.<\/p>\r\nAs the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in the table below.\r\n<table id=\"Table_03_07_003\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]0.1[\/latex]<\/td>\r\n<td>[latex]0.01[\/latex]<\/td>\r\n<td>[latex]0.001[\/latex]<\/td>\r\n<td>[latex]0.0001[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] <\/strong><\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<td>[latex]1000[\/latex]<\/td>\r\n<td>[latex]10,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe write in arrow notation:\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to {0}^{+}, f\\left(x\\right)\\to \\infty [\/latex].<\/p>\r\n<p style=\"text-align: center;\">As [latex]x[\/latex] approaches [latex]0[\/latex] from the right (positive) side, [latex]f(x)[\/latex] will approach infinity.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213909\/CNX_Precalc_Figure_03_07_0022.jpg\" alt=\"Graph of f(x)=1\/x which denotes the end behavior. As x goes to negative infinity, f(x) goes to 0, and as x goes to 0^-, f(x) goes to negative infinity. As x goes to positive infinity, f(x) goes to 0, and as x goes to 0^+, f(x) goes to positive infinity.\" width=\"731\" height=\"474\" \/>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Use arrow notation to describe the behavior of [latex]f(x)[\/latex]:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>As [latex]x \\to -1^-[\/latex]<\/li>\r\n \t<li>As [latex]x \\to -1^+[\/latex]<\/li>\r\n \t<li>As [latex]x \\to -\\infty[\/latex]<\/li>\r\n \t<li>As [latex]x \\to \\infty[\/latex]<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213918\/CNX_Precalc_Figure_03_07_0062.jpg\" alt=\"Graph of f(x)=1\/(x-2)+4 with its vertical asymptote at x=2 and its horizontal asymptote at y=4.\" width=\"487\" height=\"477\" \/><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"444547\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"444547\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>As [latex]x \\to -1^-[\/latex], [latex]f(x) \\to\u00a0 -\\infty[\/latex]<\/li>\r\n \t<li>As [latex]x \\to -1^+[\/latex], [latex]f(x) \\to \\infty[\/latex]<\/li>\r\n \t<li>As [latex]x \\to -\\infty[\/latex], [latex]f(x) \\to 4[\/latex]<\/li>\r\n \t<li>As [latex]x \\to \\infty[\/latex], [latex]f(x) \\to 4[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318940[\/ohm_question]<\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"text-align: justify;\">Utilize arrow notation to specify the behavior of rational functions.<\/li>\n<li style=\"text-align: justify;\">Find the domains of rational functions.<\/li>\n<li style=\"text-align: justify;\">Identify vertical and horizontal asymptotes.<\/li>\n<li style=\"text-align: justify;\">Identify slant asymptotes.<\/li>\n<li style=\"text-align: justify;\">Graph rational functions.<\/li>\n<\/ul>\n<\/section>\n<h2>Rational Functions<\/h2>\n<p>A <strong>rational function<\/strong> is a function that can be written as the quotient of two polynomial functions [latex]P\\left(x\\right) \\text{ and } Q\\left(x\\right)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[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]<\/p>\n<p>When a variable is present in the denominator of a rational expression, certain values of the variable may cause the denominator to equal zero. A rational expression with a zero in the denominator is not defined since we cannot divide by zero.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">We have seen the graphs of the basic <strong>reciprocal function<\/strong> and the squared reciprocal function from our study of toolkit functions.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213907\/CNX_Precalc_Figure_03_07_0012.jpg\" alt=\"Graphs of f(x)=1\/x and f(x)=1\/x^2\" width=\"731\" height=\"453\" \/>Several things are apparent if we examine the graph of [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex].<\/p>\n<ol>\n<li>On the left branch of the graph, the curve approaches the [latex]x[\/latex]-axis [latex]\\left(y=0\\right) \\text{ as } x\\to -\\infty[\/latex].<\/li>\n<li>As the graph approaches [latex]x=0[\/latex] from the left, the curve drops, but as we approach zero from the right, the curve rises.<\/li>\n<li>Finally, on the right branch of the graph, the curves approaches the [latex]x[\/latex]<em>&#8211;<\/em>axis [latex]\\left(y=0\\right) \\text{ as } x\\to \\infty[\/latex].<\/li>\n<\/ol>\n<\/section>\n<h3>Arrow Notations<\/h3>\n<p>We use <strong>arrow notation<\/strong> to show that [latex]x[\/latex]\u00a0or [latex]f\\left(x\\right)[\/latex] is approaching a particular value.<\/p>\n<table style=\"height: 240px; width: 100%;\">\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"text-align: center; height: 15px; width: 99.0867%;\" colspan=\"2\">Arrow Notation<\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<th style=\"text-align: center; height: 15px; width: 32.9933%;\">Symbol<\/th>\n<th style=\"text-align: center; height: 15px; width: 66.0934%;\">Meaning<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to {a}^{-}[\/latex]<\/td>\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches [latex]a[\/latex] from the left ([latex]x \\lt a[\/latex] but close to [latex]a[\/latex])<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to {a}^{+}[\/latex]<\/td>\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches [latex]a[\/latex] from the right ([latex]x \\lt a[\/latex] but close to [latex]a[\/latex])<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to \\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches infinity ([latex]x[\/latex]\u00a0increases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]x\\to -\\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">[latex]x[\/latex] approaches negative infinity ([latex]x[\/latex]\u00a0decreases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]f\\left(x\\right)\\to \\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">the output approaches infinity (the output increases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]f\\left(x\\right)\\to -\\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">the output approaches negative infinity (the output decreases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 32.9933%; text-align: center;\">[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\n<td style=\"height: 30px; width: 66.0934%; text-align: center;\">the output approaches [latex]a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\" aria-label=\"Example\">Let\u2019s begin by looking at the reciprocal function, [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex].As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in the table below.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]\u20130.1[\/latex]<\/td>\n<td>[latex]\u20130.01[\/latex]<\/td>\n<td>[latex]\u20130.001[\/latex]<\/td>\n<td>[latex]\u20130.0001[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] <\/strong><\/td>\n<td>[latex]\u201310[\/latex]<\/td>\n<td>[latex]\u2013100[\/latex]<\/td>\n<td>[latex]\u20131000[\/latex]<\/td>\n<td>[latex]\u201310,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We write in arrow notation:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{as }x\\to {0}^{-},f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<p style=\"text-align: center;\">As [latex]x[\/latex] approaches [latex]0[\/latex] from the left (negative) side, [latex]f(x)[\/latex] will approach negative infinity.<\/p>\n<p>As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in the table below.<\/p>\n<table id=\"Table_03_07_003\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]0.1[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]0.001[\/latex]<\/td>\n<td>[latex]0.0001[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] <\/strong><\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<td>[latex]1000[\/latex]<\/td>\n<td>[latex]10,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We write in arrow notation:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to {0}^{+}, f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n<p style=\"text-align: center;\">As [latex]x[\/latex] approaches [latex]0[\/latex] from the right (positive) side, [latex]f(x)[\/latex] will approach infinity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213909\/CNX_Precalc_Figure_03_07_0022.jpg\" alt=\"Graph of f(x)=1\/x which denotes the end behavior. As x goes to negative infinity, f(x) goes to 0, and as x goes to 0^-, f(x) goes to negative infinity. As x goes to positive infinity, f(x) goes to 0, and as x goes to 0^+, f(x) goes to positive infinity.\" width=\"731\" height=\"474\" \/><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use arrow notation to describe the behavior of [latex]f(x)[\/latex]:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>As [latex]x \\to -1^-[\/latex]<\/li>\n<li>As [latex]x \\to -1^+[\/latex]<\/li>\n<li>As [latex]x \\to -\\infty[\/latex]<\/li>\n<li>As [latex]x \\to \\infty[\/latex]<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213918\/CNX_Precalc_Figure_03_07_0062.jpg\" alt=\"Graph of f(x)=1\/(x-2)+4 with its vertical asymptote at x=2 and its horizontal asymptote at y=4.\" width=\"487\" height=\"477\" \/><\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q444547\">Show Solution<\/button><\/p>\n<div id=\"q444547\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>As [latex]x \\to -1^-[\/latex], [latex]f(x) \\to\u00a0 -\\infty[\/latex]<\/li>\n<li>As [latex]x \\to -1^+[\/latex], [latex]f(x) \\to \\infty[\/latex]<\/li>\n<li>As [latex]x \\to -\\infty[\/latex], [latex]f(x) \\to 4[\/latex]<\/li>\n<li>As [latex]x \\to \\infty[\/latex], [latex]f(x) \\to 4[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318940\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318940&theme=lumen&iframe_resize_id=ohm318940&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/84"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/84\/revisions"}],"predecessor-version":[{"id":5349,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/84\/revisions\/5349"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/84\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=84"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=84"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=84"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=84"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}