{"id":236604,"date":"2025-06-16T11:01:32","date_gmt":"2025-06-16T11:01:32","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=236604"},"modified":"2025-06-16T11:01:34","modified_gmt":"2025-06-16T11:01:34","slug":"fx-x3-25x-b-find-the-domain-and-range-of-f","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/16\/fx-x3-25x-b-find-the-domain-and-range-of-f\/","title":{"rendered":"f(x) = x^3 – 25x (b) Find the domain and range of f."},"content":{"rendered":"\n

f(x) = x^3 – 25x (b) Find the domain and range of f. (Enter your answers using interval notation.) Domain: (-\u00e2\u02c6\u017e, \u00e2\u02c6\u017e) Range: (-\u00e2\u02c6\u017e, \u00e2\u02c6\u017e) (c) State approximately the intervals on which f is increasing and on which f is decreasing. (Enter your answers using interval notation. Round your answers to two decimal places.) Increasing: (-\u00e2\u02c6\u017e, -2.92) \u00e2\u02c6\u00aa (2.92, \u00e2\u02c6\u017e) Decreasing: (-2.92, 2.92)<\/p>\n\n\n\n

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

Let us analyze the given function:f(x)=x3\u221225xf(x) = x^3 – 25xf(x)=x3\u221225x<\/p>\n\n\n\n

Part (b): Domain and Range<\/h3>\n\n\n\n

Domain:<\/strong>
The function is a polynomial. Polynomials are defined for all real numbers because there are no restrictions such as division by zero or taking the square root of a negative number. Therefore, the domain is all real numbers.Domain: (\u2212\u221e,\u221e)\\text{Domain: } (-\\infty, \\infty)Domain: (\u2212\u221e,\u221e)<\/p>\n\n\n\n

Range:<\/strong>
Similarly, the range of a cubic function that includes both odd and linear powers will also cover all real numbers. As xxx approaches positive or negative infinity, f(x)f(x)f(x) also approaches positive or negative infinity respectively.Range: (\u2212\u221e,\u221e)\\text{Range: } (-\\infty, \\infty)Range: (\u2212\u221e,\u221e)<\/p>\n\n\n\n

Part (c): Intervals of Increase and Decrease<\/h3>\n\n\n\n

To find where the function is increasing or decreasing, the first derivative is used. The first derivative represents the slope of the tangent line to the function.f\u2032(x)=3×2\u221225f'(x) = 3x^2 – 25f\u2032(x)=3×2\u221225<\/p>\n\n\n\n

Set the derivative equal to zero to find the critical points:3×2\u221225=03x^2 – 25 = 03×2\u221225=0<\/p>\n\n\n\n

Solving for xxx:x2=253x^2 = \\frac{25}{3}x2=325\u200bx=\u00b1253\u2248\u00b12.89x = \\pm \\sqrt{\\frac{25}{3}} \\approx \\pm 2.89x=\u00b1325\u200b\u200b\u2248\u00b12.89<\/p>\n\n\n\n

Now, test the intervals determined by these critical points:<\/p>\n\n\n\n

    \n
  1. For x<\u22122.89x < -2.89x<\u22122.89, choose x=\u22123x = -3x=\u22123:<\/li>\n<\/ol>\n\n\n\n

    f\u2032(\u22123)=3(\u22123)2\u221225=27\u221225=2\u2009(positive)f'(-3) = 3(-3)^2 – 25 = 27 – 25 = 2 \\, (positive)f\u2032(\u22123)=3(\u22123)2\u221225=27\u221225=2(positive)<\/p>\n\n\n\n

    The function is increasing on (\u2212\u221e,\u22122.89)(-\\infty, -2.89)(\u2212\u221e,\u22122.89).<\/p>\n\n\n\n

      \n
    1. For \u22122.89<x<2.89-2.89 < x < 2.89\u22122.89<x<2.89, choose x=0x = 0x=0:<\/li>\n<\/ol>\n\n\n\n

      f\u2032(0)=\u221225\u2009(negative)f'(0) = -25 \\, (negative)f\u2032(0)=\u221225(negative)<\/p>\n\n\n\n

      The function is decreasing on (\u22122.89,2.89)(-2.89, 2.89)(\u22122.89,2.89).<\/p>\n\n\n\n

        \n
      1. For x>2.89x > 2.89x>2.89, choose x=3x = 3x=3:<\/li>\n<\/ol>\n\n\n\n

        f\u2032(3)=3(3)2\u221225=27\u221225=2\u2009(positive)f'(3) = 3(3)^2 – 25 = 27 – 25 = 2 \\, (positive)f\u2032(3)=3(3)2\u221225=27\u221225=2(positive)<\/p>\n\n\n\n

        The function is increasing on (2.89,\u221e)(2.89, \\infty)(2.89,\u221e).<\/p>\n\n\n\n

        Correct Answer (rounded to two decimal places):<\/h3>\n\n\n\n

        Increasing:<\/strong>(\u2212\u221e,\u22122.89)\u222a(2.89,\u221e)(-\\infty, -2.89) \\cup (2.89, \\infty)(\u2212\u221e,\u22122.89)\u222a(2.89,\u221e)<\/p>\n\n\n\n

        Decreasing:<\/strong>(\u22122.89,2.89)(-2.89, 2.89)(\u22122.89,2.89)<\/p>\n\n\n\n

        This analysis shows that the function has one local maximum and one local minimum. The function increases before and after the critical points and decreases between them.<\/p>\n\n\n\n

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

        f(x) = x^3 – 25x (b) Find the domain and range of f. (Enter your answers using interval notation.) Domain: (-\u00e2\u02c6\u017e, \u00e2\u02c6\u017e) Range: (-\u00e2\u02c6\u017e, \u00e2\u02c6\u017e) (c) State approximately the intervals on which f is increasing and on which f is decreasing. (Enter your answers using interval notation. Round your answers to two decimal places.) Increasing: […]<\/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-236604","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236604","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=236604"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236604\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=236604"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=236604"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=236604"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}