{"id":1407,"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=1407"},"modified":"2015-11-12T18:35:29","modified_gmt":"2015-11-12T18:35:29","slug":"introduction-to-rational-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/introduction-to-rational-functions\/","title":{"raw":"Introduction to Rational Functions","rendered":"Introduction to Rational 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>Use arrow notation.<\/li>\n\t<li>Solve applied problems involving rational functions.<\/li>\n\t<li>Find the domains of rational functions.<\/li>\n\t<li>Identify vertical asymptotes.<\/li>\n\t<li>Identify horizontal asymptotes.<\/li>\n\t<li>Graph rational functions.<\/li>\n<\/ul><\/div>\n<p id=\"fs-id1165137740975\">Suppose we know that the cost of making a product is dependent on the number of items, <em>x<\/em>, produced. This is given by the equation [latex]C\\left(x\\right)=15,000x - 0.1{x}^{2}+1000\\\\[\/latex]. If we want to know the average cost for producing <em>x<\/em>\u00a0items, we would divide the cost function by the number of items, <em>x<\/em>.<\/p>\n<p id=\"fs-id1165137768452\">The average cost function, which yields the average cost per item for <em>x<\/em>\u00a0items produced, is<\/p>\n\n<div id=\"eip-634\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]f\\left(x\\right)=\\frac{15,000x - 0.1{x}^{2}+1000}{x}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137863708\">Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.<\/p>\n<p id=\"fs-id1165133305345\">In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.<\/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>Use arrow notation.<\/li>\n<li>Solve applied problems involving rational functions.<\/li>\n<li>Find the domains of rational functions.<\/li>\n<li>Identify vertical asymptotes.<\/li>\n<li>Identify horizontal asymptotes.<\/li>\n<li>Graph rational functions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137740975\">Suppose we know that the cost of making a product is dependent on the number of items, <em>x<\/em>, produced. This is given by the equation [latex]C\\left(x\\right)=15,000x - 0.1{x}^{2}+1000\\\\[\/latex]. If we want to know the average cost for producing <em>x<\/em>\u00a0items, we would divide the cost function by the number of items, <em>x<\/em>.<\/p>\n<p id=\"fs-id1165137768452\">The average cost function, which yields the average cost per item for <em>x<\/em>\u00a0items produced, is<\/p>\n<div id=\"eip-634\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]f\\left(x\\right)=\\frac{15,000x - 0.1{x}^{2}+1000}{x}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137863708\">Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.<\/p>\n<p id=\"fs-id1165133305345\">In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.<\/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-1407\">\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-1407","chapter","type-chapter","status-publish","hentry"],"part":1406,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1407","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1407\/revisions"}],"predecessor-version":[{"id":2377,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1407\/revisions\/2377"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1406"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1407\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/media?parent=1407"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1407"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1407"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/license?post=1407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}