{"id":3106,"date":"2025-08-15T22:01:08","date_gmt":"2025-08-15T22:01:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3106"},"modified":"2025-08-25T20:00:18","modified_gmt":"2025-08-25T20:00:18","slug":"inverse-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-functions-apply-it\/","title":{"raw":"Inverse Functions: Apply It","rendered":"Inverse Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use composition to check if two functions are inverses<\/li>\r\n \t<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one<\/li>\r\n \t<li>Find or calculate the inverse of a function<\/li>\r\n \t<li>Use the graph of a function to draw its inverse<\/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\">Shoe Size Conversions<\/h2>\r\n<p class=\"whitespace-normal break-words\">Maya works at an international shoe store and constantly needs to convert between US women's sizes and European sizes. She knows the conversion pattern:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">US size 5 \u2192 European size 35<\/li>\r\n \t<li class=\"whitespace-normal break-words\">US size 6 \u2192 European size 36<\/li>\r\n \t<li class=\"whitespace-normal break-words\">US size 7 \u2192 European size 37<\/li>\r\n \t<li class=\"whitespace-normal break-words\">US size 8 \u2192 European size 38<\/li>\r\n \t<li class=\"whitespace-normal break-words\">US size 9 \u2192 European size 39<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Maya notices that to convert from US to European, she adds 30: [latex]f(x)=x+30[\/latex] where [latex]x[\/latex] is the US size.<\/p>\r\n<p class=\"whitespace-normal break-words\">But customers also ask \"What US size is European 37?\" Now Maya needs to work backwards. She needs the inverse function.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">An <strong>inverse function<\/strong>, written as [latex]f^{-1}(x)[\/latex], reverses what the original function [latex]f(x)[\/latex] does.<\/p>\r\n<p class=\"whitespace-normal break-words\">Properties: [latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex]<\/p>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">For Maya's shoe conversions:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = x + 30[\/latex] converts US to European<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f^{-1}(x) = x - 30[\/latex] converts European to US<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]310288[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]310289[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"h-8\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]310294[\/ohm_question]\r\n\r\n<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use composition to check if two functions are inverses<\/li>\n<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one<\/li>\n<li>Find or calculate the inverse of a function<\/li>\n<li>Use the graph of a function to draw its inverse<\/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\">Shoe Size Conversions<\/h2>\n<p class=\"whitespace-normal break-words\">Maya works at an international shoe store and constantly needs to convert between US women&#8217;s sizes and European sizes. She knows the conversion pattern:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">US size 5 \u2192 European size 35<\/li>\n<li class=\"whitespace-normal break-words\">US size 6 \u2192 European size 36<\/li>\n<li class=\"whitespace-normal break-words\">US size 7 \u2192 European size 37<\/li>\n<li class=\"whitespace-normal break-words\">US size 8 \u2192 European size 38<\/li>\n<li class=\"whitespace-normal break-words\">US size 9 \u2192 European size 39<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Maya notices that to convert from US to European, she adds 30: [latex]f(x)=x+30[\/latex] where [latex]x[\/latex] is the US size.<\/p>\n<p class=\"whitespace-normal break-words\">But customers also ask &#8220;What US size is European 37?&#8221; Now Maya needs to work backwards. She needs the inverse function.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">An <strong>inverse function<\/strong>, written as [latex]f^{-1}(x)[\/latex], reverses what the original function [latex]f(x)[\/latex] does.<\/p>\n<p class=\"whitespace-normal break-words\">Properties: [latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex]<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">For Maya&#8217;s shoe conversions:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = x + 30[\/latex] converts US to European<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f^{-1}(x) = x - 30[\/latex] converts European to US<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm310288\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=310288&theme=lumen&iframe_resize_id=ohm310288&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm310289\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=310289&theme=lumen&iframe_resize_id=ohm310289&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"h-8\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm310294\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=310294&theme=lumen&iframe_resize_id=ohm310294&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":26,"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":"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\/3106"}],"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\/3106\/revisions"}],"predecessor-version":[{"id":3452,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3106\/revisions\/3452"}],"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\/3106\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3106"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3106"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3106"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3106"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}