{"id":1888,"date":"2016-11-02T22:00:36","date_gmt":"2016-11-02T22:00:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1888"},"modified":"2017-04-19T18:55:43","modified_gmt":"2017-04-19T18:55:43","slug":"end-behavior-of-rational-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/end-behavior-of-rational-functions\/","title":{"raw":"Characteristics of Rational Functions","rendered":"Characteristics of Rational Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use arrow notation to describe local and end behavior of rational functions<\/li>\r\n \t<li>Identify horizontal and vertical asymptotes of rational functions from graphs<\/li>\r\n \t<li>Graph a rational function given horizontal and vertical shifts<\/li>\r\n \t<li>Write a rational function that describes mixing<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe have seen the graphs of the basic <strong>reciprocal function<\/strong> and the squared reciprocal function from our study of toolkit functions. Examine these graphs\u00a0and notice some of their features.\r\n\r\n<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\" data-media-type=\"image\/jpg\" \/>\r\n\r\nSeveral things are apparent if we examine the graph of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex].\r\n<ol>\r\n \t<li>On the left branch of the graph, the curve approaches the <em data-effect=\"italics\">x<\/em>-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 <em data-effect=\"italics\">x-<\/em>axis [latex]\\left(y=0\\right) \\text{ as } x\\to \\infty [\/latex].<\/li>\r\n<\/ol>\r\nTo summarize, we use <strong>arrow notation<\/strong> to show that <em>x<\/em>\u00a0or [latex]f\\left(x\\right)[\/latex] is approaching a particular value.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Arrow Notation<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"text-align: center;\">Symbol<\/th>\r\n<th style=\"text-align: center;\">Meaning<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x\\to {a}^{-}[\/latex]<\/td>\r\n<td><em>x<\/em>\u00a0approaches <i>a<\/i>\u00a0from the left (<em>x\u00a0<\/em>&lt; <em>a<\/em>\u00a0but close to <em>a<\/em>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x\\to {a}^{+}[\/latex]<\/td>\r\n<td><em>x<\/em>\u00a0approaches <em>a<\/em>\u00a0from the right (<em>x\u00a0<\/em>&gt; <em>a<\/em>\u00a0but close to <em>a<\/em>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x\\to \\infty[\/latex]<\/td>\r\n<td><em>x<\/em>\u00a0approaches infinity (<em>x<\/em>\u00a0increases without bound)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x\\to -\\infty [\/latex]<\/td>\r\n<td><em>x<\/em>\u00a0approaches negative infinity (<em>x<\/em>\u00a0decreases without bound)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)\\to \\infty [\/latex]<\/td>\r\n<td>the output approaches infinity (the output increases without bound)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)\\to -\\infty [\/latex]<\/td>\r\n<td>the output approaches negative infinity (the output decreases without bound)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\r\n<td>the output approaches\u00a0<em>a<\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Local Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\r\nLet\u2019s begin by looking at the reciprocal function, [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]. We cannot divide by zero, which means the function is undefined at [latex]x=0[\/latex]; so zero is not in the domain<em data-effect=\"italics\">.<\/em> 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 id=\"Table_03_07_002\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20130.1<\/td>\r\n<td>\u20130.01<\/td>\r\n<td>\u20130.001<\/td>\r\n<td>\u20130.0001<\/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>\u201310<\/td>\r\n<td>\u2013100<\/td>\r\n<td>\u20131000<\/td>\r\n<td>\u201310,000<\/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\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><em><strong>x<\/strong><\/em><\/td>\r\n<td>0.1<\/td>\r\n<td>0.01<\/td>\r\n<td>0.001<\/td>\r\n<td>0.0001<\/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>10<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<td>10,000<\/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<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\" data-media-type=\"image\/jpg\" \/>\r\n\r\nThis behavior creates a <strong>vertical asymptote<\/strong>, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line <em>x\u00a0<\/em>= 0 as the input becomes close to zero.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213912\/CNX_Precalc_Figure_03_07_0032.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0.\" width=\"487\" height=\"364\" data-media-type=\"image\/jpg\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Vertical Asymptote<\/h3>\r\nA <strong>vertical asymptote<\/strong> of a graph is a vertical line [latex]x=a[\/latex] where the graph tends toward positive or negative infinity as the inputs approach <em>a<\/em>. We write\r\n\r\n[latex]\\text{As }x\\to a,f\\left(x\\right)\\to \\infty , \\text{or as }x\\to a,f\\left(x\\right)\\to -\\infty [\/latex].\r\n\r\n<\/div>\r\n<h2>End Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\r\nAs the values of <em>x<\/em>\u00a0approach infinity, the function values approach 0. As the values of <em>x<\/em>\u00a0approach negative infinity, the function values approach 0. Symbolically, using arrow notation\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to 0,\\text{and as }x\\to -\\infty ,f\\left(x\\right)\\to 0[\/latex].<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213913\/CNX_Precalc_Figure_03_07_0042.jpg\" alt=\"Graph of f(x)=1\/x which highlights the segments of the turning points to denote their end behavior.\" width=\"731\" height=\"475\" data-media-type=\"image\/jpg\" \/>\r\n\r\nBased on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a <strong>horizontal asymptote<\/strong>, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line [latex]y=0[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213915\/CNX_Precalc_Figure_03_07_0052.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0 and its horizontal asymptote at y=0.\" width=\"487\" height=\"364\" data-media-type=\"image\/jpg\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Horizontal Asymptote<\/h3>\r\nA <strong>horizontal asymptote<\/strong> of a graph is a horizontal line [latex]y=b[\/latex] where the graph approaches the line as the inputs increase or decrease without bound. We write\r\n\r\n[latex]\\text{As }x\\to \\infty \\text{ or }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to b[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Arrow Notation<\/h3>\r\nUse arrow notation to describe the end behavior and local behavior of the function below.\r\n\r\n<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\" data-media-type=\"image\/jpg\" \/>\r\n\r\n[reveal-answer q=\"444547\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"444547\"]\r\nNotice that the graph is showing a vertical asymptote at [latex]x=2[\/latex], which tells us that the function is undefined at [latex]x=2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to {2}^{-},f\\left(x\\right)\\to -\\infty ,\\text{ and as }x\\to {2}^{+},\\text{ }f\\left(x\\right)\\to \\infty [\/latex].<\/p>\r\nAnd as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at [latex]y=4[\/latex]. As the inputs increase without bound, the graph levels off at 4.\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to 4\\text{ and as }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to 4[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse arrow notation to describe the end behavior and local behavior for the reciprocal squared function.\r\n\r\n[reveal-answer q=\"785291\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"785291\"]End behavior: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to 0[\/latex]; Local behavior: as [latex]x\\to 0, f\\left(x\\right)\\to \\infty [\/latex] (there are no <em data-effect=\"italics\">x<\/em>- or <em data-effect=\"italics\">y<\/em>-intercepts)[\/hidden-answer]\r\n<iframe id=\"mom25\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=129042&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Transformations to Graph a Rational Function<\/h3>\r\nSketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.\r\n\r\n[reveal-answer q=\"670271\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"670271\"]\r\nShifting the graph left 2 and up 3 would result in the function\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{1}{x+2}+3[\/latex]<\/p>\r\nor equivalently, by giving the terms a common denominator,\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{3x+7}{x+2}[\/latex]<\/p>\r\nThe graph of the shifted function is displayed\u00a0below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213922\/CNX_Precalc_Figure_03_07_0072.jpg\" alt=\"Graph of f(x)=1\/(x+2)+3 with its vertical asymptote at x=-2 and its horizontal asymptote at y=3.\" width=\"731\" height=\"441\" data-media-type=\"image\/jpg\" \/>\r\n\r\nNotice that this function is undefined at [latex]x=-2[\/latex], and the graph also is showing a vertical asymptote at [latex]x=-2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to -{2}^{-}, f\\left(x\\right)\\to -\\infty ,\\text{ and as} x\\to -{2}^{+}, f\\left(x\\right)\\to \\infty [\/latex].<\/p>\r\nAs the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating a horizontal asymptote at [latex]y=3[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\pm \\infty , f\\left(x\\right)\\to 3[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nNotice that horizontal and vertical asymptotes are shifted left 2 and up 3 along with the function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\r\nSketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.\r\n\r\n[reveal-answer q=\"600482\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"600482\"]\r\n\r\nThe function and the asymptotes are shifted 3 units right and 4 units down. As [latex]x\\to 3,f\\left(x\\right)\\to \\infty[\/latex], and as [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to -4[\/latex].\r\n<p id=\"fs-id1165137823960\">The function is [latex]f\\left(x\\right)=\\frac{1}{{\\left(x - 3\\right)}^{2}}-4[\/latex].<\/p>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02000148\/CNX_Precalc_Figure_03_07_0082.jpg\"><img class=\"aligncenter size-full wp-image-2938\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02000148\/CNX_Precalc_Figure_03_07_0082.jpg\" alt=\"Graph of f(x)=1\/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.\" width=\"487\" height=\"365\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>A Mixing Problem<\/h2>\r\nIn the previous example, we shifted a toolkit function in a way that resulted in the function [latex]f\\left(x\\right)=\\frac{3x+7}{x+2}[\/latex]. This is an example of a rational function. A <strong>rational function<\/strong> is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Rational Function<\/h3>\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<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{P\\left(x\\right)}{Q\\left(x\\right)}=\\frac{{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\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Applied Problem Involving a Rational Function<\/h3>\r\nA large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning?\r\n\r\n[reveal-answer q=\"527196\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"527196\"]\r\nLet <em>t<\/em>\u00a0be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. We can write an equation independently for each:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{water: }W\\left(t\\right)=100+10t\\text{ in gallons}\\\\ \\text{sugar: }S\\left(t\\right)=5+1t\\text{ in pounds}\\end{array}[\/latex]<\/p>\r\nThe concentration, <em>C<\/em>, will be the ratio of pounds of sugar to gallons of water\r\n<p style=\"text-align: center;\">[latex]C\\left(t\\right)=\\frac{5+t}{100+10t}[\/latex]<\/p>\r\nThe concentration after 12 minutes is given by evaluating [latex]C\\left(t\\right)[\/latex] at [latex]t=\\text{ }12[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}C\\left(12\\right)=\\frac{5+12}{100+10\\left(12\\right)}\\hfill \\\\ \\text{ }=\\frac{17}{220}\\hfill \\end{array}[\/latex]<\/p>\r\nThis means the concentration is 17 pounds of sugar to 220 gallons of water.\r\n\r\nAt the beginning, the concentration is\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}C\\left(0\\right)=\\frac{5+0}{100+10\\left(0\\right)}\\hfill \\\\ \\text{ }=\\frac{1}{20}\\hfill \\end{array}[\/latex]<\/p>\r\nSince [latex]\\frac{17}{220}\\approx 0.08&gt;\\frac{1}{20}=0.05[\/latex], the concentration is greater after 12 minutes than at the beginning.\r\n<h4>Analysis of the Solution<\/h4>\r\nTo find the horizontal asymptote, divide the leading coefficient in the numerator by the leading coefficient in the denominator:\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{10}=0.1[\/latex]<\/p>\r\nNotice the horizontal asymptote is [latex]y=\\text{ }0.1[\/latex]. This means the concentration, <em>C<\/em>, the ratio of pounds of sugar to gallons of water, will approach 0.1 in the long term.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThere are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Find the ratio of freshmen to sophomores at 1 p.m.\r\n\r\n[reveal-answer q=\"526334\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"526334\"][latex]\\frac{12}{11}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=129067&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use arrow notation to describe local and end behavior of rational functions<\/li>\n<li>Identify horizontal and vertical asymptotes of rational functions from graphs<\/li>\n<li>Graph a rational function given horizontal and vertical shifts<\/li>\n<li>Write a rational function that describes mixing<\/li>\n<\/ul>\n<\/div>\n<p>We have seen the graphs of the basic <strong>reciprocal function<\/strong> and the squared reciprocal function from our study of toolkit functions. Examine these graphs\u00a0and notice some of their features.<\/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\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>Several things are apparent if we examine the graph of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex].<\/p>\n<ol>\n<li>On the left branch of the graph, the curve approaches the <em data-effect=\"italics\">x<\/em>-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 <em data-effect=\"italics\">x-<\/em>axis [latex]\\left(y=0\\right) \\text{ as } x\\to \\infty[\/latex].<\/li>\n<\/ol>\n<p>To summarize, we use <strong>arrow notation<\/strong> to show that <em>x<\/em>\u00a0or [latex]f\\left(x\\right)[\/latex] is approaching a particular value.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Arrow Notation<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Symbol<\/th>\n<th style=\"text-align: center;\">Meaning<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]x\\to {a}^{-}[\/latex]<\/td>\n<td><em>x<\/em>\u00a0approaches <i>a<\/i>\u00a0from the left (<em>x\u00a0<\/em>&lt; <em>a<\/em>\u00a0but close to <em>a<\/em>)<\/td>\n<\/tr>\n<tr>\n<td>[latex]x\\to {a}^{+}[\/latex]<\/td>\n<td><em>x<\/em>\u00a0approaches <em>a<\/em>\u00a0from the right (<em>x\u00a0<\/em>&gt; <em>a<\/em>\u00a0but close to <em>a<\/em>)<\/td>\n<\/tr>\n<tr>\n<td>[latex]x\\to \\infty[\/latex]<\/td>\n<td><em>x<\/em>\u00a0approaches infinity (<em>x<\/em>\u00a0increases without bound)<\/td>\n<\/tr>\n<tr>\n<td>[latex]x\\to -\\infty[\/latex]<\/td>\n<td><em>x<\/em>\u00a0approaches negative infinity (<em>x<\/em>\u00a0decreases without bound)<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)\\to \\infty[\/latex]<\/td>\n<td>the output approaches infinity (the output increases without bound)<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)\\to -\\infty[\/latex]<\/td>\n<td>the output approaches negative infinity (the output decreases without bound)<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\n<td>the output approaches\u00a0<em>a<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Local Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\n<p>Let\u2019s begin by looking at the reciprocal function, [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]. We cannot divide by zero, which means the function is undefined at [latex]x=0[\/latex]; so zero is not in the domain<em data-effect=\"italics\">.<\/em> 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 id=\"Table_03_07_002\" summary=\"..\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20130.1<\/td>\n<td>\u20130.01<\/td>\n<td>\u20130.001<\/td>\n<td>\u20130.0001<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] <\/strong><\/td>\n<td>\u201310<\/td>\n<td>\u2013100<\/td>\n<td>\u20131000<\/td>\n<td>\u201310,000<\/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>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><em><strong>x<\/strong><\/em><\/td>\n<td>0.1<\/td>\n<td>0.01<\/td>\n<td>0.001<\/td>\n<td>0.0001<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] <\/strong><\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<td>10,000<\/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><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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>This behavior creates a <strong>vertical asymptote<\/strong>, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line <em>x\u00a0<\/em>= 0 as the input becomes close to zero.<\/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\/02213912\/CNX_Precalc_Figure_03_07_0032.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0.\" width=\"487\" height=\"364\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Vertical Asymptote<\/h3>\n<p>A <strong>vertical asymptote<\/strong> of a graph is a vertical line [latex]x=a[\/latex] where the graph tends toward positive or negative infinity as the inputs approach <em>a<\/em>. We write<\/p>\n<p>[latex]\\text{As }x\\to a,f\\left(x\\right)\\to \\infty , \\text{or as }x\\to a,f\\left(x\\right)\\to -\\infty[\/latex].<\/p>\n<\/div>\n<h2>End Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\n<p>As the values of <em>x<\/em>\u00a0approach infinity, the function values approach 0. As the values of <em>x<\/em>\u00a0approach negative infinity, the function values approach 0. Symbolically, using arrow notation<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to 0,\\text{and as }x\\to -\\infty ,f\\left(x\\right)\\to 0[\/latex].<\/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\/02213913\/CNX_Precalc_Figure_03_07_0042.jpg\" alt=\"Graph of f(x)=1\/x which highlights the segments of the turning points to denote their end behavior.\" width=\"731\" height=\"475\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a <strong>horizontal asymptote<\/strong>, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line [latex]y=0[\/latex].<\/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\/02213915\/CNX_Precalc_Figure_03_07_0052.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0 and its horizontal asymptote at y=0.\" width=\"487\" height=\"364\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Horizontal Asymptote<\/h3>\n<p>A <strong>horizontal asymptote<\/strong> of a graph is a horizontal line [latex]y=b[\/latex] where the graph approaches the line as the inputs increase or decrease without bound. We write<\/p>\n<p>[latex]\\text{As }x\\to \\infty \\text{ or }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to b[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Arrow Notation<\/h3>\n<p>Use arrow notation to describe the end behavior and local behavior of the function below.<\/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\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q444547\">Solution<\/span><\/p>\n<div id=\"q444547\" class=\"hidden-answer\" style=\"display: none\">\nNotice that the graph is showing a vertical asymptote at [latex]x=2[\/latex], which tells us that the function is undefined at [latex]x=2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to {2}^{-},f\\left(x\\right)\\to -\\infty ,\\text{ and as }x\\to {2}^{+},\\text{ }f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n<p>And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at [latex]y=4[\/latex]. As the inputs increase without bound, the graph levels off at 4.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to 4\\text{ and as }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to 4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q785291\">Solution<\/span><\/p>\n<div id=\"q785291\" class=\"hidden-answer\" style=\"display: none\">End behavior: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to 0[\/latex]; Local behavior: as [latex]x\\to 0, f\\left(x\\right)\\to \\infty[\/latex] (there are no <em data-effect=\"italics\">x<\/em>&#8211; or <em data-effect=\"italics\">y<\/em>-intercepts)<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom25\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=129042&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Transformations to Graph a Rational Function<\/h3>\n<p>Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q670271\">Solution<\/span><\/p>\n<div id=\"q670271\" class=\"hidden-answer\" style=\"display: none\">\nShifting the graph left 2 and up 3 would result in the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{1}{x+2}+3[\/latex]<\/p>\n<p>or equivalently, by giving the terms a common denominator,<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{3x+7}{x+2}[\/latex]<\/p>\n<p>The graph of the shifted function is displayed\u00a0below.<\/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\/02213922\/CNX_Precalc_Figure_03_07_0072.jpg\" alt=\"Graph of f(x)=1\/(x+2)+3 with its vertical asymptote at x=-2 and its horizontal asymptote at y=3.\" width=\"731\" height=\"441\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>Notice that this function is undefined at [latex]x=-2[\/latex], and the graph also is showing a vertical asymptote at [latex]x=-2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to -{2}^{-}, f\\left(x\\right)\\to -\\infty ,\\text{ and as} x\\to -{2}^{+}, f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n<p>As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating a horizontal asymptote at [latex]y=3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\pm \\infty , f\\left(x\\right)\\to 3[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that horizontal and vertical asymptotes are shifted left 2 and up 3 along with the function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><br \/>\nSketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q600482\">Solution<\/span><\/p>\n<div id=\"q600482\" class=\"hidden-answer\" style=\"display: none\">\n<p>The function and the asymptotes are shifted 3 units right and 4 units down. As [latex]x\\to 3,f\\left(x\\right)\\to \\infty[\/latex], and as [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to -4[\/latex].<\/p>\n<p id=\"fs-id1165137823960\">The function is [latex]f\\left(x\\right)=\\frac{1}{{\\left(x - 3\\right)}^{2}}-4[\/latex].<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02000148\/CNX_Precalc_Figure_03_07_0082.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2938\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02000148\/CNX_Precalc_Figure_03_07_0082.jpg\" alt=\"Graph of f(x)=1\/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.\" width=\"487\" height=\"365\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>A Mixing Problem<\/h2>\n<p>In the previous example, we shifted a toolkit function in a way that resulted in the function [latex]f\\left(x\\right)=\\frac{3x+7}{x+2}[\/latex]. This is an example of a rational function. A <strong>rational function<\/strong> is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Rational Function<\/h3>\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 style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{P\\left(x\\right)}{Q\\left(x\\right)}=\\frac{{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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Applied Problem Involving a Rational Function<\/h3>\n<p>A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q527196\">Solution<\/span><\/p>\n<div id=\"q527196\" class=\"hidden-answer\" style=\"display: none\">\nLet <em>t<\/em>\u00a0be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. We can write an equation independently for each:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{water: }W\\left(t\\right)=100+10t\\text{ in gallons}\\\\ \\text{sugar: }S\\left(t\\right)=5+1t\\text{ in pounds}\\end{array}[\/latex]<\/p>\n<p>The concentration, <em>C<\/em>, will be the ratio of pounds of sugar to gallons of water<\/p>\n<p style=\"text-align: center;\">[latex]C\\left(t\\right)=\\frac{5+t}{100+10t}[\/latex]<\/p>\n<p>The concentration after 12 minutes is given by evaluating [latex]C\\left(t\\right)[\/latex] at [latex]t=\\text{ }12[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}C\\left(12\\right)=\\frac{5+12}{100+10\\left(12\\right)}\\hfill \\\\ \\text{ }=\\frac{17}{220}\\hfill \\end{array}[\/latex]<\/p>\n<p>This means the concentration is 17 pounds of sugar to 220 gallons of water.<\/p>\n<p>At the beginning, the concentration is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}C\\left(0\\right)=\\frac{5+0}{100+10\\left(0\\right)}\\hfill \\\\ \\text{ }=\\frac{1}{20}\\hfill \\end{array}[\/latex]<\/p>\n<p>Since [latex]\\frac{17}{220}\\approx 0.08>\\frac{1}{20}=0.05[\/latex], the concentration is greater after 12 minutes than at the beginning.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>To find the horizontal asymptote, divide the leading coefficient in the numerator by the leading coefficient in the denominator:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{10}=0.1[\/latex]<\/p>\n<p>Notice the horizontal asymptote is [latex]y=\\text{ }0.1[\/latex]. This means the concentration, <em>C<\/em>, the ratio of pounds of sugar to gallons of water, will approach 0.1 in the long term.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>There are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Find the ratio of freshmen to sophomores at 1 p.m.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q526334\">Solution<\/span><\/p>\n<div id=\"q526334\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{12}{11}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=129067&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\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-1888\">\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>Question ID 129042, 129067. <strong>Authored by<\/strong>: Day, Alyson. <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>: IMathAS Community License CC-BY + GPL<\/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":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 129042, 129067\",\"author\":\"Day, Alyson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"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 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