{"id":1461,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1461"},"modified":"2015-11-12T18:35:29","modified_gmt":"2015-11-12T18:35:29","slug":"introduction-to-inverses-and-radical-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/introduction-to-inverses-and-radical-functions\/","title":{"raw":"Introduction to Inverses and Radical Functions","rendered":"Introduction to Inverses and Radical Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\nBy the end of this lesson, you will be able to:\n<ul><li>Find the inverse of a polynomial function.<\/li>\n\t<li>Restrict the domain to find the inverse of a polynomial function.<\/li>\n<\/ul><\/div>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201711\/CNX_Precalc_Figure_03_08_0012.jpg\" alt=\"Gravel in the shape of a cone.\" width=\"487\" height=\"410\" data-media-type=\"image\/jpg\"\/><b>Figure 1<\/b>[\/caption]\n<p id=\"fs-id1165137793975\">A mound of gravel is in the shape of a cone with the height equal to twice the radius.<span id=\"fs-id1165137939558\" data-type=\"media\" data-alt=\"Gravel in the shape of a cone.\">\n<\/span><\/p>\n<p id=\"fs-id1165137411369\">The volume is found using a formula from elementary geometry.<\/p>\n\n<div id=\"eip-854\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}V=\\frac{1}{3}\\pi {r}^{2}h\\hfill \\\\ \\text{ }=\\frac{1}{3}\\pi {r}^{2}\\left(2r\\right)\\hfill \\\\ \\text{ }=\\frac{2}{3}\\pi {r}^{3}\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137727278\">We have written the volume <em>V<\/em>\u00a0in terms of the radius <em>r<\/em>. However, in some cases, we may start out with the volume and want to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound with a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the formula<\/p>\n\n<div id=\"eip-931\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\sqrt[3]{\\frac{3V}{2\\pi }\\\\}[\/latex]<\/div>\n<p id=\"fs-id1165134129769\">This function is the inverse of the formula for <em>V<\/em>\u00a0in terms of <em>r<\/em>.<\/p>\n<p id=\"fs-id1165137656509\">In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.<\/p>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\n<p>By the end of this lesson, you will be able to:<\/p>\n<ul>\n<li>Find the inverse of a polynomial function.<\/li>\n<li>Restrict the domain to find the inverse of a polynomial function.<\/li>\n<\/ul>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201711\/CNX_Precalc_Figure_03_08_0012.jpg\" alt=\"Gravel in the shape of a cone.\" width=\"487\" height=\"410\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137793975\">A mound of gravel is in the shape of a cone with the height equal to twice the radius.<span id=\"fs-id1165137939558\" data-type=\"media\" data-alt=\"Gravel in the shape of a cone.\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137411369\">The volume is found using a formula from elementary geometry.<\/p>\n<div id=\"eip-854\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}V=\\frac{1}{3}\\pi {r}^{2}h\\hfill \\\\ \\text{ }=\\frac{1}{3}\\pi {r}^{2}\\left(2r\\right)\\hfill \\\\ \\text{ }=\\frac{2}{3}\\pi {r}^{3}\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137727278\">We have written the volume <em>V<\/em>\u00a0in terms of the radius <em>r<\/em>. However, in some cases, we may start out with the volume and want to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound with a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the formula<\/p>\n<div id=\"eip-931\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]r=\\sqrt[3]{\\frac{3V}{2\\pi }\\\\}[\/latex]<\/div>\n<p id=\"fs-id1165134129769\">This function is the inverse of the formula for <em>V<\/em>\u00a0in terms of <em>r<\/em>.<\/p>\n<p id=\"fs-id1165137656509\">In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1461\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1461","chapter","type-chapter","status-publish","hentry"],"part":1459,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1461","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1461\/revisions"}],"predecessor-version":[{"id":2367,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1461\/revisions\/2367"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1459"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1461\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1461"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1461"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1461"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}