{"id":1994,"date":"2024-06-27T00:44:03","date_gmt":"2024-06-27T00:44:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1994"},"modified":"2025-08-15T13:44:23","modified_gmt":"2025-08-15T13:44:23","slug":"module-10-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-10-background-youll-need-1\/","title":{"raw":"Rational and Radical Functions: Background You'll Need 1","rendered":"Rational and Radical Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Figure out which values make a rational expression impossible to calculate (like dividing by zero)<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"font-600 text-xl font-bold\">Rational Expressions and Undefined Values<\/h2>\r\n<p class=\"whitespace-pre-wrap break-words\">A rational expression is an algebraic expression that can be written as the quotient of two polynomials, [latex]P(x)[\/latex] and [latex]Q(x)[\/latex], where [latex]Q(x) \\neq 0[\/latex]. It takes the general form:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{P(x)}{Q(x)}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">where [latex]P(x)[\/latex] is the numerator and [latex]Q(x)[\/latex] is the denominator.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The domain of a rational expression includes all real numbers except those that make the denominator equal to zero. When the denominator equals zero, the expression is undefined. This concept is rooted in the fundamental principle that division by zero is impossible in mathematics.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>rational expressions and undefined values<\/h3>\r\nA rational expression is undefined when its denominator equals zero.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the undefined values of a rational expression<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Isolate the denominator.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set the denominator equal to zero.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation.<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">The solutions to this equation are the values that make the rational expression undefined.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine the value(s) of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 - 4}{x - 2}[\/latex]<\/p>\r\n[reveal-answer q=\"138302\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"138302\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x - 2)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x - 2 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:\r\n[latex]x = 2[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Note: At [latex]x = 2[\/latex], both the numerator and denominator equal zero, creating an [pb_glossary id=\"4395\"]indeterminate form[\/pb_glossary] [latex]\\frac{0}{0}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the values of [latex]x[\/latex] that make the following rational expression undefined:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 + 3x}{x^2 - 4}[\/latex]<\/p>\r\n[reveal-answer q=\"298475\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"298475\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x^2 - 4)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x^2 - 4 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor the equation: [latex](x + 2)(x - 2) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:\r\n[latex]x + 2 = 0[\/latex] or [latex]x - 2 = 0[\/latex]\r\n[latex]x = -2[\/latex] or [latex]x = 2[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex] or [latex]x = -2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine all values of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{2x^2 - 5}{x^3 - x}[\/latex]<\/p>\r\n[reveal-answer q=\"156506\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"156506\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the denominator: [latex]x^3 - x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor the denominator: [latex]x(x^2 - 1) = x(x + 1)(x - 1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set each factor to zero and solve:\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">\u00a0[latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x + 1 = 0[\/latex], so [latex]x = -1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x - 1 = 0[\/latex], so [latex]x = 1[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 0[\/latex], [latex]x = 1[\/latex], and [latex]x = -1[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>Undefined values in rational expressions correspond to vertical asymptotes in their graphs. As [latex]x[\/latex] approaches an undefined value, the expression approaches infinity or negative infinity, creating a vertical line that the graph approaches but never crosses.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">For example, in the graph of [latex]y = \\frac{1}{x - 2}[\/latex]:<\/p>\r\n\r\n\r\n[caption id=\"attachment_4394\" align=\"aligncenter\" width=\"867\"]<img class=\"wp-image-4394 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805.png\" alt=\"Graph of y = \\frac{1}{x - 2}\" width=\"867\" height=\"833\" \/> Graph of y = 1 \/ (x-2)[\/caption]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the left, [latex]y[\/latex] approaches positive infinity.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the right, [latex]y[\/latex] approaches negative infinity.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The line [latex]x = 2[\/latex] is a vertical asymptote of the graph.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"font-600 text-xl font-bold\"><strong>Common Mistakes to Avoid<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Forgetting to check for undefined values before simplifying.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Assuming only linear terms in the denominator can cause undefined values.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Neglecting to factor completely when dealing with higher degree polynomials.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Confusing zero values of the numerator with undefined values.<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Figure out which values make a rational expression impossible to calculate (like dividing by zero)<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 class=\"font-600 text-xl font-bold\">Rational Expressions and Undefined Values<\/h2>\n<p class=\"whitespace-pre-wrap break-words\">A rational expression is an algebraic expression that can be written as the quotient of two polynomials, [latex]P(x)[\/latex] and [latex]Q(x)[\/latex], where [latex]Q(x) \\neq 0[\/latex]. It takes the general form:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{P(x)}{Q(x)}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">where [latex]P(x)[\/latex] is the numerator and [latex]Q(x)[\/latex] is the denominator.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The domain of a rational expression includes all real numbers except those that make the denominator equal to zero. When the denominator equals zero, the expression is undefined. This concept is rooted in the fundamental principle that division by zero is impossible in mathematics.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>rational expressions and undefined values<\/h3>\n<p>A rational expression is undefined when its denominator equals zero.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the undefined values of a rational expression<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Isolate the denominator.<\/li>\n<li class=\"whitespace-normal break-words\">Set the denominator equal to zero.<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation.<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">The solutions to this equation are the values that make the rational expression undefined.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine the value(s) of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 - 4}{x - 2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q138302\">Show Answer<\/button><\/p>\n<div id=\"q138302\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x - 2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x - 2 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:<br \/>\n[latex]x = 2[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Note: At [latex]x = 2[\/latex], both the numerator and denominator equal zero, creating an <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1994_4395\">indeterminate form<\/a> [latex]\\frac{0}{0}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the values of [latex]x[\/latex] that make the following rational expression undefined:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 + 3x}{x^2 - 4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q298475\">Show Answer<\/button><\/p>\n<div id=\"q298475\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x^2 - 4)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x^2 - 4 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor the equation: [latex](x + 2)(x - 2) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:<br \/>\n[latex]x + 2 = 0[\/latex] or [latex]x - 2 = 0[\/latex]<br \/>\n[latex]x = -2[\/latex] or [latex]x = 2[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex] or [latex]x = -2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine all values of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{2x^2 - 5}{x^3 - x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q156506\">Show Answer<\/button><\/p>\n<div id=\"q156506\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the denominator: [latex]x^3 - x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor the denominator: [latex]x(x^2 - 1) = x(x + 1)(x - 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set each factor to zero and solve:\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">\u00a0[latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x + 1 = 0[\/latex], so [latex]x = -1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x - 1 = 0[\/latex], so [latex]x = 1[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 0[\/latex], [latex]x = 1[\/latex], and [latex]x = -1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Undefined values in rational expressions correspond to vertical asymptotes in their graphs. As [latex]x[\/latex] approaches an undefined value, the expression approaches infinity or negative infinity, creating a vertical line that the graph approaches but never crosses.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">For example, in the graph of [latex]y = \\frac{1}{x - 2}[\/latex]:<\/p>\n<figure id=\"attachment_4394\" aria-describedby=\"caption-attachment-4394\" style=\"width: 867px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4394 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805.png\" alt=\"Graph of y = \\frac{1}{x - 2}\" width=\"867\" height=\"833\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805.png 867w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805-300x288.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805-768x738.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805-65x62.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805-225x216.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805-350x336.png 350w\" sizes=\"(max-width: 867px) 100vw, 867px\" \/><figcaption id=\"caption-attachment-4394\" class=\"wp-caption-text\">Graph of y = 1 \/ (x-2)<\/figcaption><\/figure>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the left, [latex]y[\/latex] approaches positive infinity.<\/li>\n<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the right, [latex]y[\/latex] approaches negative infinity.<\/li>\n<li class=\"whitespace-normal break-words\">The line [latex]x = 2[\/latex] is a vertical asymptote of the graph.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"font-600 text-xl font-bold\"><strong>Common Mistakes to Avoid<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Forgetting to check for undefined values before simplifying.<\/li>\n<li class=\"whitespace-normal break-words\">Assuming only linear terms in the denominator can cause undefined values.<\/li>\n<li class=\"whitespace-normal break-words\">Neglecting to factor completely when dealing with higher degree polynomials.<\/li>\n<li class=\"whitespace-normal break-words\">Confusing zero values of the numerator with undefined values.<\/li>\n<\/ol>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_1994_4395\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1994_4395\"><div tabindex=\"-1\"><p>An indeterminate form occurs when a mathematical expression yields an ambiguous result. The form 0\/0 is one such case.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":12,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":232,"module-header":"background_you_need","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1994"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1994\/revisions"}],"predecessor-version":[{"id":7778,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1994\/revisions\/7778"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/232"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1994\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1994"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1994"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1994"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1994"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}