{"id":195835,"date":"2025-03-01T07:57:10","date_gmt":"2025-03-01T07:57:10","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195835"},"modified":"2025-03-01T07:57:13","modified_gmt":"2025-03-01T07:57:13","slug":"discuss-the-validity-of-each-statement","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/01\/discuss-the-validity-of-each-statement\/","title":{"rendered":"Discuss the validity of each statement"},"content":{"rendered":"\n<p>Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. 62. Every polynomial function is a rational function. 63. Every rational function is a polynomial function. 64. The graph of every rational function has at least one vertical asymptote. 65. The graph of every exponential function has a horizontal asymptote. 66. The graph of every logarithmic function has a vertical asymptote. 67. There exists a rational function that has both a vertical and horizontal asymptote.<\/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>Let&#8217;s analyze each of these statements one by one, providing an explanation and counterexamples where applicable.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">62. Every polynomial function is a rational function.<\/h3>\n\n\n\n<p><strong>True<\/strong>. A polynomial function is any function of the form:<\/p>\n\n\n\n<p>[<br>P(x) = a_n x^n + a_{n-1} x^{n-1} + \\dots + a_1 x + a_0<br>]<\/p>\n\n\n\n<p>This is a sum of terms where the exponents of (x) are non-negative integers. A rational function is any function that can be expressed as a ratio of two polynomials, i.e.,<\/p>\n\n\n\n<p>[<br>R(x) = \\frac{P(x)}{Q(x)}<br>]<\/p>\n\n\n\n<p>where (P(x)) and (Q(x)) are polynomials, and (Q(x) \\neq 0). Every polynomial can be written as a rational function by expressing it as:<\/p>\n\n\n\n<p>[<br>\\frac{P(x)}{1}<br>]<\/p>\n\n\n\n<p>Since 1 is a polynomial and non-zero, every polynomial is technically a rational function.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">63. Every rational function is a polynomial function.<\/h3>\n\n\n\n<p><strong>False<\/strong>. A rational function is a ratio of two polynomials. It is not necessarily a polynomial function. For example:<\/p>\n\n\n\n<p>[<br>R(x) = \\frac{1}{x}<br>]<\/p>\n\n\n\n<p>This is a rational function but not a polynomial because it has a denominator. Polynomials do not have denominators with variables. Hence, not every rational function is a polynomial.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">64. The graph of every rational function has at least one vertical asymptote.<\/h3>\n\n\n\n<p><strong>False<\/strong>. Not all rational functions have vertical asymptotes. A vertical asymptote occurs when the denominator of a rational function approaches zero, but the numerator does not. For example:<\/p>\n\n\n\n<p>[<br>R(x) = \\frac{x+1}{x^2 + 1}<br>]<\/p>\n\n\n\n<p>The denominator (x^2 + 1) never equals zero, so there is no vertical asymptote in this case. Thus, not every rational function has a vertical asymptote.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">65. The graph of every exponential function has a horizontal asymptote.<\/h3>\n\n\n\n<p><strong>True<\/strong>. Exponential functions generally take the form:<\/p>\n\n\n\n<p>[<br>f(x) = a \\cdot b^x<br>]<\/p>\n\n\n\n<p>where (a) is a constant and (b) is a positive real number. As (x) approaches infinity or negative infinity, the function approaches a horizontal line, typically (y = 0). This means exponential functions have a horizontal asymptote, usually at (y = 0), unless the function is transformed (e.g., shifted vertically).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">66. The graph of every logarithmic function has a vertical asymptote.<\/h3>\n\n\n\n<p><strong>True<\/strong>. Logarithmic functions take the form:<\/p>\n\n\n\n<p>[<br>f(x) = a \\cdot \\log_b(x &#8211; h) + k<br>]<\/p>\n\n\n\n<p>For (x &gt; h), the graph approaches negative infinity as (x) approaches (h), meaning there is a vertical asymptote at (x = h). This is true for all basic logarithmic functions, and the vertical asymptote occurs where the argument inside the logarithm is zero.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">67. There exists a rational function that has both a vertical and horizontal asymptote.<\/h3>\n\n\n\n<p><strong>True<\/strong>. A rational function can have both vertical and horizontal asymptotes. For example:<\/p>\n\n\n\n<p>[<br>R(x) = \\frac{1}{x^2 + 1}<br>]<\/p>\n\n\n\n<p>This function has a vertical asymptote at (x = 0) (where the denominator approaches zero) and a horizontal asymptote at (y = 0) as (x) approaches infinity. Thus, such rational functions do exist.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>62<\/strong>: True (Every polynomial is a rational function).<\/li>\n\n\n\n<li><strong>63<\/strong>: False (Not every rational function is a polynomial).<\/li>\n\n\n\n<li><strong>64<\/strong>: False (Not every rational function has a vertical asymptote).<\/li>\n\n\n\n<li><strong>65<\/strong>: True (Every exponential function has a horizontal asymptote).<\/li>\n\n\n\n<li><strong>66<\/strong>: True (Every logarithmic function has a vertical asymptote).<\/li>\n\n\n\n<li><strong>67<\/strong>: True (There exists a rational function with both vertical and horizontal asymptotes).<\/li>\n<\/ul>\n\n\n\n<p>I will generate a plot of rational functions with vertical and horizontal asymptotes for a visual understanding.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-4.png\" alt=\"\" class=\"wp-image-195836\"\/><\/figure>\n\n\n\n<p>Here is a graph showing a rational function with both a vertical and horizontal asymptote, helping to visualize the behavior of such functions. The vertical asymptote occurs at (x = 0), and the horizontal asymptote is at (y = 0). This demonstrates how rational functions can have both types of asymptotes. Let me know if you need further explanations!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. 62. Every polynomial function is a rational function. 63. Every rational function is a polynomial function. 64. The graph of every rational function has at least one vertical asymptote. 65. The graph of every exponential [&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-195835","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195835","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=195835"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195835\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195835"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195835"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195835"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}