{"id":2823,"date":"2025-08-13T18:45:08","date_gmt":"2025-08-13T18:45:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2823"},"modified":"2026-01-31T02:02:26","modified_gmt":"2026-01-31T02:02:26","slug":"introduction-to-calculus-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/introduction-to-calculus-background-youll-need-2\/","title":{"raw":"Introduction to Calculus: Background You'll Need 2","rendered":"Introduction to Calculus: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate piecewise functions given its equation<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Evaluating Piecewise Functions from Equations<\/h2>\r\nPiecewise functions can also be defined using equations that include multiple rules. Each rule applies only when its corresponding condition is met.\r\n\r\nTo evaluate a piecewise function given its equation, determine which condition the input satisfies and then use the appropriate rule.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Piecewise-defined equation<\/h3>\r\nA <b>piecewise-defined equation<\/b> lists multiple formulas, each with a condition that specifies when it should be used.\r\n\r\n<\/div>\r\nA piecewise function is commonly written in the following form:\r\n[latex]\\\\[\/latex]\r\n[latex]\r\nf(x)=\r\n\\begin{cases}\r\n\\text{expression}_1 &amp; \\text{condition}_1 \\\\\r\n\\text{expression}_2 &amp; \\text{condition}_2 \\\\\r\n\\text{expression}_3 &amp; \\text{condition}_3\r\n\\end{cases}\r\n[\/latex]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Consider the piecewise-defined function below.[latex]f(x)=\r\n\\begin{cases}\r\nx+2 &amp; x&lt;0 \\\\\r\nx^2 &amp; 0\\le x&lt;2 \\\\\r\n5 &amp; x\\ge2\r\n\\end{cases}\r\n[\/latex]\r\n\r\nEvaluate the function at each input value:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(-1)[\/latex]<\/li>\r\n \t<li>[latex]f(0)[\/latex]<\/li>\r\n \t<li>[latex]f(3)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"190587\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"190587\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>\u00a0Since [latex]-1&lt;0[\/latex], use the first rule. [latex]f(-1)=-1+2=1[\/latex]<\/li>\r\n \t<li>Since [latex]0\\leq 0&lt;2[\/latex], use the second rule. [latex]f(0)=0^2=0[\/latex]<\/li>\r\n \t<li>Since [latex]3 \\ge2[\/latex], use the third rule. [latex]f(3)=5[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]319415[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate piecewise functions given its equation<\/li>\n<\/ul>\n<\/section>\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Evaluating Piecewise Functions from Equations<\/h2>\n<p>Piecewise functions can also be defined using equations that include multiple rules. Each rule applies only when its corresponding condition is met.<\/p>\n<p>To evaluate a piecewise function given its equation, determine which condition the input satisfies and then use the appropriate rule.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Piecewise-defined equation<\/h3>\n<p>A <b>piecewise-defined equation<\/b> lists multiple formulas, each with a condition that specifies when it should be used.<\/p>\n<\/div>\n<p>A piecewise function is commonly written in the following form:<br \/>\n[latex]\\\\[\/latex]<br \/>\n[latex]f(x)=  \\begin{cases}  \\text{expression}_1 & \\text{condition}_1 \\\\  \\text{expression}_2 & \\text{condition}_2 \\\\  \\text{expression}_3 & \\text{condition}_3  \\end{cases}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Consider the piecewise-defined function below.[latex]f(x)=  \\begin{cases}  x+2 & x<0 \\\\  x^2 & 0\\le x<2 \\\\  5 & x\\ge2  \\end{cases}[\/latex]\n\nEvaluate the function at each input value:\n\n\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(-1)[\/latex]<\/li>\n<li>[latex]f(0)[\/latex]<\/li>\n<li>[latex]f(3)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q190587\">Show Solution<\/button><\/p>\n<div id=\"q190587\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>\u00a0Since [latex]-1<0[\/latex], use the first rule. [latex]f(-1)=-1+2=1[\/latex]<\/li>\n<li>Since [latex]0\\leq 0<2[\/latex], use the second rule. [latex]f(0)=0^2=0[\/latex]<\/li>\n<li>Since [latex]3 \\ge2[\/latex], use the third rule. [latex]f(3)=5[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm319415\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=319415&theme=lumen&iframe_resize_id=ohm319415&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2823"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2823\/revisions"}],"predecessor-version":[{"id":5472,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2823\/revisions\/5472"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2823\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2823"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2823"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2823"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2823"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}