{"id":255696,"date":"2025-07-13T07:35:46","date_gmt":"2025-07-13T07:35:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=255696"},"modified":"2025-07-13T07:35:48","modified_gmt":"2025-07-13T07:35:48","slug":"which-of-the-following-is-the-domain-and-range-of","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/13\/which-of-the-following-is-the-domain-and-range-of\/","title":{"rendered":"Which of the following is the domain and range of\u00a0"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The correct answer is:
Domain: (-\u221e, \u221e), Range: (-\u03c0\/2, \u03c0\/2)<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

To understand the domain and range of the inverse tangent function, y = tan\u207b\u00b9(x), it is helpful to first consider the original tangent function, y = tan(x).<\/p>\n\n\n\n

    \n
  1. The Tangent Function,\u00a0y = tan(x):<\/strong>\n
      \n
    • The tangent function takes an angle (in radians) as its input and gives a ratio as its output.<\/li>\n\n\n\n
    • The\u00a0range<\/strong>\u00a0of the tangent function is all real numbers. This means it can produce any value from negative infinity to positive infinity, written as\u00a0(-\u221e, \u221e).<\/li>\n\n\n\n
    • The\u00a0domain<\/strong>\u00a0of the tangent function is not all real numbers because it has vertical asymptotes at odd multiples of\u00a0\u03c0\/2\u00a0(e.g.,\u00a0… -\u03c0\/2, \u03c0\/2, 3\u03c0\/2, …).<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • Defining the Inverse Tangent Function,\u00a0y = tan\u207b\u00b9(x):<\/strong>\n
        \n
      • For a function to have an inverse, it must be one-to-one, meaning each output value corresponds to only one input value. The tangent function is periodic and not one-to-one over its entire domain.<\/li>\n\n\n\n
      • To define an inverse, we restrict the domain of\u00a0tan(x)\u00a0to a principal branch where it is one-to-one. The standard choice for this restricted domain is the open interval\u00a0(-\u03c0\/2, \u03c0\/2). On this interval, the\u00a0tan(x)\u00a0function covers its entire range of\u00a0(-\u221e, \u221e).<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Swapping Domain and Range for the Inverse:<\/strong>\n
          \n
        • The inverse function,\u00a0y = tan\u207b\u00b9(x), essentially swaps the roles of the input and output from the original (restricted) function.<\/li>\n\n\n\n
        • The\u00a0domain<\/strong>\u00a0of\u00a0tan\u207b\u00b9(x)\u00a0is the\u00a0range<\/strong>\u00a0of the restricted\u00a0tan(x). Since the range of\u00a0tan(x)\u00a0is\u00a0(-\u221e, \u221e), the domain of\u00a0tan\u207b\u00b9(x)\u00a0is also\u00a0(-\u221e, \u221e). This means you can take the inverse tangent of any real number.<\/li>\n\n\n\n
        • The\u00a0range<\/strong>\u00a0of\u00a0tan\u207b\u00b9(x)\u00a0is the\u00a0domain<\/strong>\u00a0of the restricted\u00a0tan(x). Since the restricted domain of\u00a0tan(x)\u00a0is\u00a0(-\u03c0\/2, \u03c0\/2), the range of\u00a0tan\u207b\u00b9(x)\u00a0is also\u00a0(-\u03c0\/2, \u03c0\/2).<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

          The parentheses in the range (-\u03c0\/2, \u03c0\/2) are crucial. They indicate an open interval, meaning the output values get infinitely close to -\u03c0\/2 and \u03c0\/2 but never actually reach them. This is because the graph of y = tan\u207b\u00b9(x) has horizontal asymptotes at y = -\u03c0\/2 and y = \u03c0\/2.<\/p>\n\n\n\n

          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          The Correct Answer and Explanation is: The correct answer is:Domain: (-\u221e, \u221e), Range: (-\u03c0\/2, \u03c0\/2) Explanation: To understand the domain and range of the inverse tangent function, y = tan\u207b\u00b9(x), it is helpful to first consider the original tangent function, y = tan(x). The parentheses in the range (-\u03c0\/2, \u03c0\/2) are crucial. They indicate an open interval, meaning the output values […]<\/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-255696","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/255696","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=255696"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/255696\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=255696"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=255696"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=255696"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}