{"id":3146,"date":"2025-08-15T22:18:03","date_gmt":"2025-08-15T22:18:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3146"},"modified":"2026-01-14T19:14:35","modified_gmt":"2026-01-14T19:14:35","slug":"rational-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-apply-it\/","title":{"raw":"Rational Functions: Apply It","rendered":"Rational Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Utilize arrow notation to specify the behavior of rational functions.<\/li>\r\n \t<li>Find the domains of rational functions.<\/li>\r\n \t<li>Identify vertical and horizontal asymptotes.<\/li>\r\n \t<li>Identify slant asymptotes.<\/li>\r\n \t<li>Graph rational functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Physics - Optical Lens Systems<\/h2>\r\n<p class=\"whitespace-normal break-words\">A physics lab is studying how light behaves through different lens systems. When two converging lenses are placed at varying distances apart, the effective focal length of the compound system changes dramatically. The effective focal length [latex]f(d)[\/latex] (in cm) depends on the distance [latex]d[\/latex] (in cm) between the lenses according to:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]f(d) = \\frac{12d}{d^2 - 16}[\/latex]<\/p>\r\n\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Optical System Analysis<\/h3>\r\n<p class=\"whitespace-normal break-words\">Let's analyze this lens system to understand how distance affects focusing behavior and identify critical separation distances.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Consider the function: [latex]f(d) = \\frac{12d}{d^2 - 16}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Determine where the lens system is undefined.<\/p>\r\n<p class=\"whitespace-normal break-words\">[reveal-answer q=\"572279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"572279\"]To find where the function is undefined, we set the denominator equal to zero: [latex]\\begin{aligned} d^2 - 16 &amp;= 0 \\ d^2 &amp;= 16 \\ d &amp;= \\pm 4 \\end{aligned}[\/latex] The function is undefined when [latex]d = 4[\/latex] cm and [latex]d = -4[\/latex] cm. Interpretation: Since distance between lenses must be positive, we only consider [latex]d = 4[\/latex] cm. At this separation distance, the optical system cannot produce a well-defined focal length - this represents a critical configuration where the lens system fails to focus properly.[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Again, consider the function: [latex]f(d) = \\frac{12d}{d^2 - 16}[\/latex].Analyze the long-distance behavior of the lens system.<\/p>\r\n[reveal-answer q=\"250668\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"250668\"]For the horizontal asymptote, we compare the degrees of the numerator and denominator: Degree of numerator: 1 Degree of denominator: 2 Since the denominator has a higher degree, the horizontal asymptote is [latex]y = 0[\/latex]. Interpretation: As the lenses are placed very far apart ([latex]d \\to \\infty[\/latex]), the effective focal length approaches zero. This means the compound lens system loses its ability to focus light effectively when the lenses are too far separated.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311226[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Utilize arrow notation to specify the behavior of rational functions.<\/li>\n<li>Find the domains of rational functions.<\/li>\n<li>Identify vertical and horizontal asymptotes.<\/li>\n<li>Identify slant asymptotes.<\/li>\n<li>Graph rational functions.<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Physics &#8211; Optical Lens Systems<\/h2>\n<p class=\"whitespace-normal break-words\">A physics lab is studying how light behaves through different lens systems. When two converging lenses are placed at varying distances apart, the effective focal length of the compound system changes dramatically. The effective focal length [latex]f(d)[\/latex] (in cm) depends on the distance [latex]d[\/latex] (in cm) between the lenses according to:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]f(d) = \\frac{12d}{d^2 - 16}[\/latex]<\/p>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Optical System Analysis<\/h3>\n<p class=\"whitespace-normal break-words\">Let&#8217;s analyze this lens system to understand how distance affects focusing behavior and identify critical separation distances.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Consider the function: [latex]f(d) = \\frac{12d}{d^2 - 16}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Determine where the lens system is undefined.<\/p>\n<p class=\"whitespace-normal break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q572279\">Show Solution<\/button><\/p>\n<div id=\"q572279\" class=\"hidden-answer\" style=\"display: none\">To find where the function is undefined, we set the denominator equal to zero: [latex]\\begin{aligned} d^2 - 16 &= 0 \\ d^2 &= 16 \\ d &= \\pm 4 \\end{aligned}[\/latex] The function is undefined when [latex]d = 4[\/latex] cm and [latex]d = -4[\/latex] cm. Interpretation: Since distance between lenses must be positive, we only consider [latex]d = 4[\/latex] cm. At this separation distance, the optical system cannot produce a well-defined focal length &#8211; this represents a critical configuration where the lens system fails to focus properly.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Again, consider the function: [latex]f(d) = \\frac{12d}{d^2 - 16}[\/latex].Analyze the long-distance behavior of the lens system.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q250668\">Show Solution<\/button><\/p>\n<div id=\"q250668\" class=\"hidden-answer\" style=\"display: none\">For the horizontal asymptote, we compare the degrees of the numerator and denominator: Degree of numerator: 1 Degree of denominator: 2 Since the denominator has a higher degree, the horizontal asymptote is [latex]y = 0[\/latex]. Interpretation: As the lenses are placed very far apart ([latex]d \\to \\infty[\/latex]), the effective focal length approaches zero. This means the compound lens system loses its ability to focus light effectively when the lenses are too far separated.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311226\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311226&theme=lumen&iframe_resize_id=ohm311226&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"apply_it","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3146"}],"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\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3146\/revisions"}],"predecessor-version":[{"id":5356,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3146\/revisions\/5356"}],"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\/3146\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3146"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3146"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3146"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}