{"id":168703,"date":"2024-11-18T09:01:09","date_gmt":"2024-11-18T09:01:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=168703"},"modified":"2024-11-18T09:01:12","modified_gmt":"2024-11-18T09:01:12","slug":"find-the-domain-range-and-sketch-the-graph-of-the-function-fx-2%e2%88%92x-4x%e2%88%9216-through-the-asymptotes-and-the-xy-intercept-points","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/18\/find-the-domain-range-and-sketch-the-graph-of-the-function-fx-2%e2%88%92x-4x%e2%88%9216-through-the-asymptotes-and-the-xy-intercept-points\/","title":{"rendered":"Find the domain, range and sketch the graph of the function f(x)= [2\u2212x]\/{4x\u221216] through the asymptotes and the x,y-intercept points."},"content":{"rendered":"\n

Find the domain, range and sketch the graph of the function f(x)= [2\u2212x]\/{4x\u221216] through the asymptotes and the x,y-intercept points.<\/p>\n\n\n\n

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

To analyze the function f(x)=2\u2212x4x\u221216f(x) = \\frac{2 – x}{4x – 16}f(x)=4x\u2212162\u2212x\u200b, we will determine its domain<\/strong>, range<\/strong>, asymptotes, intercepts, and graph it.<\/p>\n\n\n\n

\n\n\n\n

Domain<\/strong>:<\/h3>\n\n\n\n

The denominator 4x\u221216=04x – 16 = 04x\u221216=0 when x=4x = 4x=4. At x=4x = 4x=4, the function is undefined.
Domain<\/strong>: x\u2208(\u2212\u221e,4)\u222a(4,\u221e)x \\in (-\\infty, 4) \\cup (4, \\infty)x\u2208(\u2212\u221e,4)\u222a(4,\u221e).<\/p>\n\n\n\n

\n\n\n\n

Intercepts<\/strong>:<\/h3>\n\n\n\n
    \n
  • x-intercept<\/strong>: The numerator equals 0 when 2\u2212x=02 – x = 02\u2212x=0, i.e., x=2x = 2x=2.
    Thus, the x-intercept is (2,0)(2, 0)(2,0).<\/li>\n\n\n\n
  • y-intercept<\/strong>: Substituting x=0x = 0x=0,f(0)=2\u221204(0)\u221216=2\u221216=\u221218.f(0) = \\frac{2 – 0}{4(0) – 16} = \\frac{2}{-16} = -\\frac{1}{8}.f(0)=4(0)\u2212162\u22120\u200b=\u2212162\u200b=\u221281\u200b.Thus, the y-intercept is (0,\u221218)(0, -\\frac{1}{8})(0,\u221281\u200b).<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Asymptotes<\/strong>:<\/h3>\n\n\n\n
      \n
    1. Vertical asymptote<\/strong>: The denominator 4x\u221216=04x – 16 = 04x\u221216=0, so the vertical asymptote is x=4x = 4x=4.<\/li>\n\n\n\n
    2. Horizontal asymptote<\/strong>: As x\u2192\u00b1\u221ex \\to \\pm\\inftyx\u2192\u00b1\u221e, the highest degree terms dominate. f(x)\u223c\u2212x4x=\u221214.f(x) \\sim \\frac{-x}{4x} = -\\frac{1}{4}.f(x)\u223c4x\u2212x\u200b=\u221241\u200b. Thus, the horizontal asymptote is y=\u221214y = -\\frac{1}{4}y=\u221241\u200b.<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      Range<\/strong>:<\/h3>\n\n\n\n

      The function approaches \u221214-\\frac{1}{4}\u221241\u200b but never equals it, and spans all other values because there are no restrictions on f(x)f(x)f(x).
      Range<\/strong>: y\u2208(\u2212\u221e,\u221214)\u222a(\u221214,\u221e)y \\in (-\\infty, -\\frac{1}{4}) \\cup (-\\frac{1}{4}, \\infty)y\u2208(\u2212\u221e,\u221241\u200b)\u222a(\u221241\u200b,\u221e).<\/p>\n\n\n\n

      \n\n\n\n

      Behavior and Sketch<\/strong>:<\/h3>\n\n\n\n
        \n
      • Near x=4x = 4x=4, f(x)f(x)f(x) diverges to \u00b1\u221e\\pm\\infty\u00b1\u221e depending on the sign of x\u22124x – 4x\u22124.<\/li>\n\n\n\n
      • For large xxx, f(x)f(x)f(x) stabilizes near y=\u221214y = -\\frac{1}{4}y=\u221241\u200b.<\/li>\n<\/ul>\n\n\n\n

        The graph shows a hyperbolic shape, with the x-intercept at (2,0)(2, 0)(2,0), y-intercept at (0,\u221218)(0, -\\frac{1}{8})(0,\u221281\u200b), and asymptotes at x=4x = 4x=4 and y=\u221214y = -\\frac{1}{4}y=\u221241\u200b.
        Would you like me to generate the graph?<\/p>\n","protected":false},"excerpt":{"rendered":"

        Find the domain, range and sketch the graph of the function f(x)= [2\u2212x]\/{4x\u221216] through the asymptotes and the x,y-intercept points. The Correct Answer and Explanation is: To analyze the function f(x)=2\u2212x4x\u221216f(x) = \\frac{2 – x}{4x – 16}f(x)=4x\u2212162\u2212x\u200b, we will determine its domain, range, asymptotes, intercepts, and graph it. Domain: The denominator 4x\u221216=04x – 16 = […]<\/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-168703","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168703","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=168703"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168703\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=168703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=168703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=168703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}