{"id":202182,"date":"2025-03-17T08:57:39","date_gmt":"2025-03-17T08:57:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=202182"},"modified":"2025-03-17T08:57:41","modified_gmt":"2025-03-17T08:57:41","slug":"which-of-the-following-statements-is-false-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/17\/which-of-the-following-statements-is-false-5\/","title":{"rendered":"Which of the following statements is FALSE"},"content":{"rendered":"\n

Which of the following statements is FALSE?<\/p>\n\n\n\n

A) All polynomial functions are continuous everywhere.<\/p>\n\n\n\n

B) The absolute value function f(x) = l x l is continuous everywhere<\/p>\n\n\n\n

C) All rational functions are continuous everywhere <\/p>\n\n\n\n

D) Square root function is always continuous at every number in its domain.<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

The false<\/strong> statement is:<\/p>\n\n\n\n

C) All rational functions are continuous everywhere.<\/strong><\/p>\n\n\n\n

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

A function is continuous<\/strong> at a point if there are no breaks, jumps, or asymptotes at that point. Now, let\u2019s analyze each statement carefully:<\/p>\n\n\n\n

    \n
  1. Statement A (True):<\/strong> Polynomial functions (e.g., ( f(x) = x^2 + 3x – 5 ), ( g(x) = x^5 – 2x^3 + 4 )) are continuous everywhere because they are built from sums, products, and powers of ( x ), which are always continuous. There are no restrictions or undefined points in polynomial functions.<\/li>\n\n\n\n
  2. Statement B (True):<\/strong> The absolute value function, ( f(x) = |x| ), is continuous everywhere<\/strong>. Even though it has a sharp corner at ( x = 0 ), it does not have any jumps or breaks, meaning the left-hand and right-hand limits exist and are equal to the function value.<\/li>\n\n\n\n
  3. Statement C (False):<\/strong> Rational functions are not always continuous everywhere<\/strong> because they have denominators that can be zero, leading to undefined points (vertical asymptotes).<\/strong> A rational function is a ratio of two polynomials:
    [
    R(x) = \\frac{P(x)}{Q(x)}
    ]
    It is continuous except where ( Q(x) = 0 ),<\/strong> which creates discontinuities (holes or vertical asymptotes). For example,
    [
    f(x) = \\frac{1}{x-3}
    ]
    is undefined at ( x = 3 ), making it discontinuous<\/strong> there.<\/li>\n\n\n\n
  4. Statement D (True):<\/strong> The square root function, ( f(x) = \\sqrt{x} ), is continuous for all values in its domain<\/strong> (i.e., ( x \\geq 0 )). There are no jumps or breaks in its valid range.<\/li>\n<\/ol>\n\n\n\n

    Thus, the incorrect (false) statement is C<\/strong>, because rational functions can have points of discontinuity.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Which of the following statements is FALSE? A) All polynomial functions are continuous everywhere. B) The absolute value function f(x) = l x l is continuous everywhere C) All rational functions are continuous everywhere D) Square root function is always continuous at every number in its domain. The correct answer and explanation is : The […]<\/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-202182","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/202182","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=202182"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/202182\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=202182"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=202182"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=202182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}