{"id":3116,"date":"2025-08-15T22:04:56","date_gmt":"2025-08-15T22:04:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3116"},"modified":"2025-12-30T20:04:18","modified_gmt":"2025-12-30T20:04:18","slug":"absolute-value-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/absolute-value-functions-apply-it\/","title":{"raw":"Absolute Value Functions: Apply It","rendered":"Absolute Value Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph an absolute value function.<\/li>\r\n \t<li>Solve an absolute value equation.<\/li>\r\n \t<li>Solve an absolute value inequality.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">Understanding absolute value functions is helpful for calculus because of the unique sharp corner.<\/p>\r\n\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Function Continuity Analysis<\/h3>\r\n<p class=\"whitespace-normal break-words\">Consider the function [latex]f(x) = |x - 2| + 1[\/latex]. In calculus, you'll need to analyze where functions are continuous and where they have breaks or corners.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-pre-wrap break-words\">[ohm_question hide_question_numbers=1]317636[\/ohm_question]<\/p>\r\n\r\n<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Converting to Piecewise Form<\/h3>\r\n<p class=\"whitespace-normal break-words\">To understand behavior around the critical point, we write [latex]f(x) = |x - 2| + 1[\/latex] as a piecewise function.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>Converting Absolute Value to Piecewise<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Find where the expression inside equals zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Test values on either side of this point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write separate expressions for each piece<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify continuity at the boundary point<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Rewrite [latex]f(x) = |x - 2| + 1[\/latex] as a piecewise function.<\/p>\r\n\r\n<ul>\r\n \t<li>For [latex]x - 2 \\geq 0[\/latex] we get a domain of [latex]x \\geq 2[\/latex].The positive piece of the absolute value:<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 40px;\">[latex]\r\n\\begin{align}\r\nf(x) &amp;= (x - 2) + 1 \\\\\r\n&amp;= x - 1\r\n\\end{align}\r\n[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li>For [latex]x - 2 &lt; 0[\/latex] we get a domain of [latex]x &lt; 2[\/latex].<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 40px;\">The negative piece of the absolute value:<\/p>\r\n<p style=\"padding-left: 40px;\">[latex]\r\n\\begin{align}\r\nf(x) &amp;= -(x - 2) + 1 \\\\\r\n&amp;= -x + 3\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Piecewise form:<\/strong> [latex]f(x) = \\begin{cases} -x + 3 &amp; \\text{if } x &lt; 2 \\\\ x - 1 &amp; \\text{if } x \\geq 2 \\end{cases}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]317637[\/ohm_question]<\/p>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph an absolute value function.<\/li>\n<li>Solve an absolute value equation.<\/li>\n<li>Solve an absolute value inequality.<\/li>\n<\/ul>\n<\/section>\n<p class=\"whitespace-normal break-words\">Understanding absolute value functions is helpful for calculus because of the unique sharp corner.<\/p>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Function Continuity Analysis<\/h3>\n<p class=\"whitespace-normal break-words\">Consider the function [latex]f(x) = |x - 2| + 1[\/latex]. In calculus, you&#8217;ll need to analyze where functions are continuous and where they have breaks or corners.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-pre-wrap break-words\"><iframe loading=\"lazy\" id=\"ohm317636\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317636&theme=lumen&iframe_resize_id=ohm317636&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Converting to Piecewise Form<\/h3>\n<p class=\"whitespace-normal break-words\">To understand behavior around the critical point, we write [latex]f(x) = |x - 2| + 1[\/latex] as a piecewise function.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>Converting Absolute Value to Piecewise<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Find where the expression inside equals zero<\/li>\n<li class=\"whitespace-normal break-words\">Test values on either side of this point<\/li>\n<li class=\"whitespace-normal break-words\">Write separate expressions for each piece<\/li>\n<li class=\"whitespace-normal break-words\">Verify continuity at the boundary point<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Rewrite [latex]f(x) = |x - 2| + 1[\/latex] as a piecewise function.<\/p>\n<ul>\n<li>For [latex]x - 2 \\geq 0[\/latex] we get a domain of [latex]x \\geq 2[\/latex].The positive piece of the absolute value:<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">[latex]\\begin{align}  f(x) &= (x - 2) + 1 \\\\  &= x - 1  \\end{align}[\/latex]<\/p>\n<ul>\n<li>For [latex]x - 2 < 0[\/latex] we get a domain of [latex]x < 2[\/latex].<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">The negative piece of the absolute value:<\/p>\n<p style=\"padding-left: 40px;\">[latex]\\begin{align}  f(x) &= -(x - 2) + 1 \\\\  &= -x + 3  \\end{align}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Piecewise form:<\/strong> [latex]f(x) = \\begin{cases} -x + 3 & \\text{if } x < 2 \\\\ x - 1 & \\text{if } x \\geq 2 \\end{cases}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm317637\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317637&theme=lumen&iframe_resize_id=ohm317637&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":34,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"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\/3116"}],"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":17,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3116\/revisions"}],"predecessor-version":[{"id":5159,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3116\/revisions\/5159"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3116\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3116"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3116"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3116"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}