{"id":233187,"date":"2025-06-12T19:19:50","date_gmt":"2025-06-12T19:19:50","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233187"},"modified":"2025-06-12T19:19:54","modified_gmt":"2025-06-12T19:19:54","slug":"which-exponential-function-has-an-initial-value-of-2-fx-23x-fx-32x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/12\/which-exponential-function-has-an-initial-value-of-2-fx-23x-fx-32x\/","title":{"rendered":"Which exponential function has an initial value of 2? f(x) = 2(3x) f(x) = 3(2x)"},"content":{"rendered":"\n<p>Which exponential function has an initial value of 2? f(x) = 2(3x) f(x) = 3(2x)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h6 class=\"wp-block-heading\">ChatGPT said:<\/h6>\n\n\n\n<p><strong>Correct Answer:<\/strong><br>The exponential function with an initial value of 2 is:<br><strong>f(x) = 2(3^x)<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Explanation:<\/strong><\/p>\n\n\n\n<p>In an exponential function, the general form is:<\/p>\n\n\n\n<p><strong>f(x) = a(b^x)<\/strong><\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>a<\/strong> is the initial value (also called the y-intercept),<\/li>\n\n\n\n<li><strong>b<\/strong> is the base or growth\/decay factor,<\/li>\n\n\n\n<li><strong>x<\/strong> is the exponent (usually the independent variable).<\/li>\n<\/ul>\n\n\n\n<p>The <strong>initial value<\/strong> is the output of the function when <strong>x = 0<\/strong>. To determine the initial value of a function, substitute <strong>x = 0<\/strong> into the equation:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>f(x) = 2(3^x)<\/strong><br>f(0) = 2(3^0) = 2(1) = <strong>2<\/strong><\/li>\n\n\n\n<li><strong>f(x) = 3(2^x)<\/strong><br>f(0) = 3(2^0) = 3(1) = <strong>3<\/strong><\/li>\n<\/ol>\n\n\n\n<p>The first function, <strong>f(x) = 2(3^x)<\/strong>, has an initial value of <strong>2<\/strong>, which matches the condition in the question. This makes it the correct exponential function.<\/p>\n\n\n\n<p>The value <strong>a<\/strong> in the expression <strong>a(b^x)<\/strong> controls the vertical stretch or compression of the function and gives the value of the function at <strong>x = 0<\/strong>. The base <strong>b<\/strong> determines the rate of exponential growth (if b &gt; 1) or decay (if 0 &lt; b &lt; 1).<\/p>\n\n\n\n<p>In the case of <strong>f(x) = 2(3^x)<\/strong>, the function starts at 2 when x is 0 and grows rapidly as x increases because the base 3 is greater than 1. This type of function models real-world phenomena such as compound interest, population growth, and bacterial reproduction, where the quantity increases by a fixed multiple over equal time intervals.<\/p>\n\n\n\n<p>Understanding how the parameters of an exponential function affect its behavior is essential in interpreting and modeling exponential change in various applications.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-203.jpeg\" alt=\"\" class=\"wp-image-233188\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Which exponential function has an initial value of 2? f(x) = 2(3x) f(x) = 3(2x) The Correct Answer and Explanation is: ChatGPT said: Correct Answer:The exponential function with an initial value of 2 is:f(x) = 2(3^x) Explanation: In an exponential function, the general form is: f(x) = a(b^x) Where: The initial value is the output [&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 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