{"id":151314,"date":"2024-10-09T19:43:21","date_gmt":"2024-10-09T19:43:21","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=151314"},"modified":"2024-10-09T19:43:24","modified_gmt":"2024-10-09T19:43:24","slug":"what-is-the-parent-function-of-a-rational-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/09\/what-is-the-parent-function-of-a-rational-function\/","title":{"rendered":"What is the parent function of a rational function"},"content":{"rendered":"\n<p>What is the parent function of a rational function?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>The parent function of a rational function is defined as <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-3-color\"><strong>( f(x) = \\frac{1}{x} )<\/strong><\/mark>. This function serves as the foundational model for all rational functions, which can be expressed as the quotient of two polynomial functions, ( f(x) = \\frac{p(x)}{q(x)} ), where ( p(x) ) and ( q(x) ) are polynomials.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Characteristics of the Parent Function ( f(x) = \\frac{1}{x} )<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Domain and Range<\/strong>: The domain of ( f(x) = \\frac{1}{x} ) excludes zero, as division by zero is undefined. Thus, the domain is ( x \\in (-\\infty, 0) \\cup (0, \\infty) ). The range is also ( y \\in (-\\infty, 0) \\cup (0, \\infty) ), meaning the function can take any positive or negative value but never zero.<\/li>\n\n\n\n<li><strong>Asymptotes<\/strong>: This function has two asymptotes:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Vertical Asymptote<\/strong>: At ( x = 0 ), where the function approaches infinity as ( x ) approaches zero from the right and negative infinity as ( x ) approaches zero from the left.<\/li>\n\n\n\n<li><strong>Horizontal Asymptote<\/strong>: As ( x ) approaches positive or negative infinity, ( f(x) ) approaches zero.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Graph<\/strong>: The graph of ( f(x) = \\frac{1}{x} ) consists of two distinct branches located in the first and third quadrants of the Cartesian plane. The curve is symmetric about the origin, demonstrating odd symmetry (i.e., ( f(-x) = -f(x) )).<\/li>\n\n\n\n<li><strong>Transformations<\/strong>: Any rational function can be derived from this parent function by applying transformations such as vertical and horizontal shifts, reflections, stretches, and compressions. For example, the function ( f(x) = \\frac{1}{x &#8211; 2} + 3 ) represents a horizontal shift to the right by 2 units and a vertical shift upward by 3 units.<\/li>\n<\/ol>\n\n\n\n<p>Overall, understanding the parent function ( f(x) = \\frac{1}{x} ) provides a solid foundation for studying more complex rational functions, enabling a better grasp of their behavior and characteristics.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is the parent function of a rational function? The Correct Answer and Explanation is : The parent function of a rational function is defined as ( f(x) = \\frac{1}{x} ). This function serves as the foundational model for all rational functions, which can be expressed as the quotient of two polynomial functions, ( f(x) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-151314","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/151314","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=151314"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/151314\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=151314"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=151314"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=151314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}