{"id":489,"date":"2025-07-10T17:43:43","date_gmt":"2025-07-10T17:43:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=489"},"modified":"2025-10-29T16:27:55","modified_gmt":"2025-10-29T16:27:55","slug":"working-with-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/working-with-functions-background-youll-need-1\/","title":{"raw":"Working with Functions: Background You'll Need 4","rendered":"Working with Functions: Background You&#8217;ll Need 4"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Rearrange equations for a variable of interest<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div class=\"bcc-box bcc-success\"><article class=\"text-token-text-primary w-full focus:outline-none [--shadow-height:45px] has-data-writing-block:pointer-events-none has-data-writing-block:-mt-(--shadow-height) has-data-writing-block:pt-(--shadow-height) [&amp;:has([data-writing-block])&gt;*]:pointer-events-auto [content-visibility:auto] supports-[content-visibility:auto]:[contain-intrinsic-size:auto_100lvh] scroll-mt-[calc(var(--header-height)+min(200px,max(70px,20svh)))]\" dir=\"auto\" tabindex=\"-1\" data-turn-id=\"request-WEB:3fb6258c-0d1b-4336-b16a-8edd80891682-4\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"true\" data-turn=\"assistant\">\r\n<div class=\"text-base my-auto mx-auto pb-10 [--thread-content-margin:--spacing(4)] thread-sm:[--thread-content-margin:--spacing(6)] thread-lg:[--thread-content-margin:--spacing(16)] px-(--thread-content-margin)\">\r\n<div class=\"[--thread-content-max-width:40rem] thread-lg:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group\/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn\" tabindex=\"-1\">\r\n<div class=\"flex max-w-full flex-col grow\">\r\n<div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&amp;]:mt-1\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"2fdcc4ed-db73-48c2-a30b-51aeedafb5d0\" data-message-model-slug=\"gpt-5\">\r\n<div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[1px]\">\r\n<div class=\"markdown prose dark:prose-invert w-full break-words light markdown-new-styling\">\r\n<p data-start=\"163\" data-end=\"382\">Let\u2019s practice rearranging equations so that one variable is isolated.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"416\" data-end=\"478\">Solve for [latex]x[\/latex]:<br data-start=\"443\" data-end=\"446\" \/>[latex]2(-6x - 8y) = -9x[\/latex]<\/p>\r\n<p data-start=\"416\" data-end=\"478\">[reveal-answer q=\"38925\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"38925\"]<\/p>\r\n<p data-start=\"480\" data-end=\"542\"><strong data-start=\"480\" data-end=\"491\">Step 1:<\/strong> Distribute the 2<br data-start=\"508\" data-end=\"511\" \/>[latex]-12x - 16y = -9x[\/latex]<\/p>\r\n<p data-start=\"544\" data-end=\"632\"><strong data-start=\"544\" data-end=\"555\">Step 2:<\/strong> Move all [latex]x[\/latex]-terms to one side<br data-start=\"599\" data-end=\"602\" \/>[latex]-12x + 9x = 16y[\/latex]<\/p>\r\n<p data-start=\"634\" data-end=\"744\"><strong data-start=\"634\" data-end=\"645\">Step 3:<\/strong> Combine like terms and divide<br data-start=\"675\" data-end=\"678\" \/>[latex]-3x = 16y \\quad\\Rightarrow\\quad x = -\\tfrac{16}{3}y[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"888\" data-end=\"944\">Solve for [latex]x[\/latex]:<br data-start=\"915\" data-end=\"918\" \/>[latex]y = x^2 + 4[\/latex]<\/p>\r\n<p data-start=\"888\" data-end=\"944\">[reveal-answer q=\"553939\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"553939\"]Step 1: Subtract 4 [latex]y - 4 = x^2[\/latex] Step 2: Take the square root of both sides [latex]x = \\pm \\sqrt{y - 4}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">When solving for [latex]x[\/latex] in a squared equation, remember the [latex]\\pm[\/latex] symbol \u2014 both positive and negative roots matter.\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]314674[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"1598\" data-end=\"1663\">Solve for [latex]x[\/latex]:<br data-start=\"1625\" data-end=\"1628\" \/>[latex]y = \\dfrac{4}{9 - x}[\/latex]<\/p>\r\n<p data-start=\"1598\" data-end=\"1663\">\r\n[reveal-answer q=\"778440\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"778440\"]<\/p>\r\n<p data-start=\"1665\" data-end=\"1752\"><strong data-start=\"1665\" data-end=\"1676\">Step 1:<\/strong> Multiply both sides by [latex](9 - x)[\/latex]<br data-start=\"1722\" data-end=\"1725\" \/>[latex]y(9 - x) = 4[\/latex]<\/p>\r\n<p data-start=\"1754\" data-end=\"1815\"><strong data-start=\"1754\" data-end=\"1765\">Step 2:<\/strong> Expand and rearrange<br data-start=\"1786\" data-end=\"1789\" \/>[latex]9y - xy = 4[\/latex]<\/p>\r\n<p data-start=\"1817\" data-end=\"1900\"><strong data-start=\"1817\" data-end=\"1828\">Step 3:<\/strong> Move terms with [latex]x[\/latex] together<br data-start=\"1870\" data-end=\"1873\" \/>[latex]-xy = 4 - 9y[\/latex]<\/p>\r\n<p data-start=\"1902\" data-end=\"1980\"><strong data-start=\"1902\" data-end=\"1913\">Step 4:<\/strong> Divide by [latex]-y[\/latex]<br data-start=\"1941\" data-end=\"1944\" \/>[latex]x = \\dfrac{9y - 4}{y}[\/latex]<\/p>\r\n<p data-start=\"1598\" data-end=\"1663\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]314675[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/article><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Rearrange equations for a variable of interest<\/li>\n<\/ul>\n<\/section>\n<div class=\"bcc-box bcc-success\">\n<article class=\"text-token-text-primary w-full focus:outline-none [--shadow-height:45px] has-data-writing-block:pointer-events-none has-data-writing-block:-mt-(--shadow-height) has-data-writing-block:pt-(--shadow-height) [&amp;:has([data-writing-block])&gt;*]:pointer-events-auto [content-visibility:auto] supports-[content-visibility:auto]:[contain-intrinsic-size:auto_100lvh] scroll-mt-[calc(var(--header-height)+min(200px,max(70px,20svh)))]\" dir=\"auto\" tabindex=\"-1\" data-turn-id=\"request-WEB:3fb6258c-0d1b-4336-b16a-8edd80891682-4\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"true\" data-turn=\"assistant\">\n<div class=\"text-base my-auto mx-auto pb-10 [--thread-content-margin:--spacing(4)] thread-sm:[--thread-content-margin:--spacing(6)] thread-lg:[--thread-content-margin:--spacing(16)] px-(--thread-content-margin)\">\n<div class=\"[--thread-content-max-width:40rem] thread-lg:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group\/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn\" tabindex=\"-1\">\n<div class=\"flex max-w-full flex-col grow\">\n<div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&amp;]:mt-1\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"2fdcc4ed-db73-48c2-a30b-51aeedafb5d0\" data-message-model-slug=\"gpt-5\">\n<div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[1px]\">\n<div class=\"markdown prose dark:prose-invert w-full break-words light markdown-new-styling\">\n<p data-start=\"163\" data-end=\"382\">Let\u2019s practice rearranging equations so that one variable is isolated.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"416\" data-end=\"478\">Solve for [latex]x[\/latex]:<br data-start=\"443\" data-end=\"446\" \/>[latex]2(-6x - 8y) = -9x[\/latex]<\/p>\n<p data-start=\"416\" data-end=\"478\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q38925\">Show Solution<\/button><\/p>\n<div id=\"q38925\" class=\"hidden-answer\" style=\"display: none\">\n<p data-start=\"480\" data-end=\"542\"><strong data-start=\"480\" data-end=\"491\">Step 1:<\/strong> Distribute the 2<br data-start=\"508\" data-end=\"511\" \/>[latex]-12x - 16y = -9x[\/latex]<\/p>\n<p data-start=\"544\" data-end=\"632\"><strong data-start=\"544\" data-end=\"555\">Step 2:<\/strong> Move all [latex]x[\/latex]-terms to one side<br data-start=\"599\" data-end=\"602\" \/>[latex]-12x + 9x = 16y[\/latex]<\/p>\n<p data-start=\"634\" data-end=\"744\"><strong data-start=\"634\" data-end=\"645\">Step 3:<\/strong> Combine like terms and divide<br data-start=\"675\" data-end=\"678\" \/>[latex]-3x = 16y \\quad\\Rightarrow\\quad x = -\\tfrac{16}{3}y[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"888\" data-end=\"944\">Solve for [latex]x[\/latex]:<br data-start=\"915\" data-end=\"918\" \/>[latex]y = x^2 + 4[\/latex]<\/p>\n<p data-start=\"888\" data-end=\"944\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q553939\">Show Solution<\/button><\/p>\n<div id=\"q553939\" class=\"hidden-answer\" style=\"display: none\">Step 1: Subtract 4 [latex]y - 4 = x^2[\/latex] Step 2: Take the square root of both sides [latex]x = \\pm \\sqrt{y - 4}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">When solving for [latex]x[\/latex] in a squared equation, remember the [latex]\\pm[\/latex] symbol \u2014 both positive and negative roots matter.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm314674\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=314674&theme=lumen&iframe_resize_id=ohm314674&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"1598\" data-end=\"1663\">Solve for [latex]x[\/latex]:<br data-start=\"1625\" data-end=\"1628\" \/>[latex]y = \\dfrac{4}{9 - x}[\/latex]<\/p>\n<p data-start=\"1598\" data-end=\"1663\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q778440\">Show Solution<\/button><\/p>\n<div id=\"q778440\" class=\"hidden-answer\" style=\"display: none\">\n<p data-start=\"1665\" data-end=\"1752\"><strong data-start=\"1665\" data-end=\"1676\">Step 1:<\/strong> Multiply both sides by [latex](9 - x)[\/latex]<br data-start=\"1722\" data-end=\"1725\" \/>[latex]y(9 - x) = 4[\/latex]<\/p>\n<p data-start=\"1754\" data-end=\"1815\"><strong data-start=\"1754\" data-end=\"1765\">Step 2:<\/strong> Expand and rearrange<br data-start=\"1786\" data-end=\"1789\" \/>[latex]9y - xy = 4[\/latex]<\/p>\n<p data-start=\"1817\" data-end=\"1900\"><strong data-start=\"1817\" data-end=\"1828\">Step 3:<\/strong> Move terms with [latex]x[\/latex] together<br data-start=\"1870\" data-end=\"1873\" \/>[latex]-xy = 4 - 9y[\/latex]<\/p>\n<p data-start=\"1902\" data-end=\"1980\"><strong data-start=\"1902\" data-end=\"1913\">Step 4:<\/strong> Divide by [latex]-y[\/latex]<br data-start=\"1941\" data-end=\"1944\" \/>[latex]x = \\dfrac{9y - 4}{y}[\/latex]<\/p>\n<p data-start=\"1598\" data-end=\"1663\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm314675\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=314675&theme=lumen&iframe_resize_id=ohm314675&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/article>\n<\/div>\n","protected":false},"author":13,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":498,"module-header":"- Select Header -","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\/489"}],"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\/13"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/489\/revisions"}],"predecessor-version":[{"id":4905,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/489\/revisions\/4905"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/498"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/489\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=489"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=489"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=489"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=489"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}