What is the multiplicative rate of change for the exponential function f(x)=2((5)\/(2))^(-x)<\/p>\n\n\n\n
Which of these following statements are true?
is
is<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n Let’s analyze and solve the question step by step.<\/p>\n\n\n\n The multiplicative rate of change<\/strong> of an exponential function is the base<\/strong> of the exponent (what the function is multiplied by each time x increases by 1).<\/p>\n\n\n\n Given the function: f(x)=2(52)\u2212xf(x) = 2\\left(\\frac{5}{2}\\right)^{-x}<\/p>\n\n\n\n We need to identify the base of the exponential expression.<\/p>\n\n\n\n (52)\u2212x=(25)x\\left(\\frac{5}{2}\\right)^{-x} = \\left(\\frac{2}{5}\\right)^x<\/p>\n\n\n\n This is because a\u2212x=1axa^{-x} = \\frac{1}{a^x}. So: f(x)=2\u22c5(25)xf(x) = 2 \\cdot \\left(\\frac{2}{5}\\right)^x<\/p>\n\n\n\n Now, we can see that the base (or the multiplicative rate of change) is: 25\\boxed{\\frac{2}{5}}<\/p>\n\n\n\n This means that for every increase of 1 in xx, the output of the function is multiplied by 25\\frac{2}{5}<\/strong>.<\/p>\n\n\n\n Since options (a) and (b) are not complete in your message, I\u2019ll assume you meant to list some statements related to the function. Let me provide two common statements based on typical exponential function analysis:<\/p>\n\n\n\n Statement (a):<\/strong> The function is increasing. To determine which is true, consider the form again: f(x)=2(25)xf(x) = 2\\left(\\frac{2}{5}\\right)^x<\/p>\n\n\n\n The base 25\\frac{2}{5} is less than 1<\/strong>, so this is a decreasing exponential function<\/strong>. As xx increases, f(x)f(x) gets smaller.<\/p>\n\n\n\n Thus, Statement (b)<\/strong> is true<\/strong>, and Statement (a)<\/strong> is false<\/strong>.<\/p>\n\n\n\n A.<\/strong> The multiplicative rate of change is 25\\boxed{\\frac{2}{5}}. Exponential functions have the general form f(x)=abxf(x) = ab^x, where:<\/p>\n\n\n\n In the problem, we\u2019re given: f(x)=2(52)\u2212xf(x) = 2\\left(\\frac{5}{2}\\right)^{-x}<\/p>\n\n\n\n We start by simplifying this expression. Recall that a negative exponent means taking the reciprocal: (52)\u2212x=(25)x\\left(\\frac{5}{2}\\right)^{-x} = \\left(\\frac{2}{5}\\right)^x<\/p>\n\n\n\n Therefore, we rewrite the function as: f(x)=2(25)xf(x) = 2\\left(\\frac{2}{5}\\right)^x<\/p>\n\n\n\n Now, the base is clearly 25\\frac{2}{5}, and that is the multiplicative rate of change. This tells us that the output of the function is multiplied by 25\\frac{2}{5}<\/strong> every time x increases by 1. Since 25\\frac{2}{5} is a value between 0 and 1, this function is decreasing<\/strong>. That means as x gets larger, f(x)f(x) gets smaller.<\/p>\n\n\n\n In part B, we’re asked which statement is true between (a) and (b), and although their content was missing, a common pair of choices involves identifying whether a function is increasing or decreasing. Because our base is 25\\frac{2}{5}, and that\u2019s less than 1, we know the function is decreasing. Therefore, the true statement is (b): “The function is decreasing.”<\/p>\n\n\n\n Understanding how the base of an exponential function controls the direction of the graph is critical in algebra and precalculus. It helps in interpreting growth and decay models in science, finance, and real-life situations.<\/p>\n","protected":false},"excerpt":{"rendered":" What is the multiplicative rate of change for the exponential function f(x)=2((5)\/(2))^(-x) Which of these following statements are true? is is The Correct Answer and Explanation is: Let’s analyze and solve the question step by step. Part A: Multiplicative Rate of Change The multiplicative rate of change of an exponential function is the base of […]<\/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-212853","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/212853","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=212853"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/212853\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=212853"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=212853"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=212853"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nPart A: Multiplicative Rate of Change<\/strong><\/h3>\n\n\n\n
Step 1: Understand the exponent<\/h4>\n\n\n\n
\n\n\n\nPart B: Which of the following statements are true?<\/strong><\/h3>\n\n\n\n
Statement (b):<\/strong> The function is decreasing.<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answers:<\/h3>\n\n\n\n
B.<\/strong> (b) is true\\boxed{\\text{(b) is true}}<\/p>\n\n\n\n
\n\n\n\n\ud83d\udcd8 Explanation (300+ words):<\/h3>\n\n\n\n
\n