{"id":243251,"date":"2025-07-04T13:14:52","date_gmt":"2025-07-04T13:14:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243251"},"modified":"2025-07-04T13:14:54","modified_gmt":"2025-07-04T13:14:54","slug":"given-the-function-fx1x5-calculate-the-following-values","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/given-the-function-fx1x5-calculate-the-following-values\/","title":{"rendered":"Given the function\u00a0f(x)=1x+5, calculate the following values"},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">&nbsp;points) Given the function&nbsp;f(x)=1x+5, calculate the following values:<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-122.png\" alt=\"\" class=\"wp-image-243254\"\/><\/figure>\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<p>Of course. Here are the correct answers and a detailed explanation for the problem.<\/p>\n\n\n\n<p><strong>Answers:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>f(a) =<\/strong>\u00a01 \/ (a + 5)<\/li>\n\n\n\n<li><strong>f(a + h) =<\/strong>\u00a01 \/ (a + h + 5)<\/li>\n\n\n\n<li><strong>(f(a + h) &#8211; f(a)) \/ h =<\/strong>\u00a0-1 \/ ((a + 5)(a + h + 5))<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This problem asks you to work with function notation and then to calculate and simplify the difference quotient, which is a fundamental concept in calculus used to define the derivative of a function.<\/p>\n\n\n\n<p><strong>1. Calculating f(a)<\/strong><\/p>\n\n\n\n<p>The first step is to evaluate the function&nbsp;f(x)&nbsp;at the value&nbsp;x = a. The original function is given as&nbsp;f(x) = 1 \/ (x + 5). To find&nbsp;f(a), we simply replace every instance of&nbsp;x&nbsp;in the function&#8217;s formula with&nbsp;a.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Original function:\u00a0f(x) = 1 \/ (x + 5)<\/li>\n\n\n\n<li>Substitute\u00a0x\u00a0with\u00a0a:\u00a0f(a) = 1 \/ (a + 5)<\/li>\n<\/ul>\n\n\n\n<p>This expression cannot be simplified further, so it is the final answer for the first part.<\/p>\n\n\n\n<p><strong>2. Calculating f(a + h)<\/strong><\/p>\n\n\n\n<p>The second step is similar, but this time we evaluate the function at&nbsp;x = a + h. We substitute the entire expression&nbsp;(a + h)&nbsp;for&nbsp;x&nbsp;in the original function.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Original function:\u00a0f(x) = 1 \/ (x + 5)<\/li>\n\n\n\n<li>Substitute\u00a0x\u00a0with\u00a0(a + h):\u00a0f(a + h) = 1 \/ ((a + h) + 5)<\/li>\n\n\n\n<li>Removing the inner parentheses gives:\u00a0f(a + h) = 1 \/ (a + h + 5)<\/li>\n<\/ul>\n\n\n\n<p>This is the final simplified expression for the second part.<\/p>\n\n\n\n<p><strong>3. Calculating the Difference Quotient: (f(a + h) &#8211; f(a)) \/ h<\/strong><\/p>\n\n\n\n<p>This final part requires you to use the results from the first two steps and perform algebraic simplification. We start by substituting the expressions for&nbsp;f(a + h)&nbsp;and&nbsp;f(a)&nbsp;into the difference quotient formula.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Start with the formula:\u00a0(f(a + h) &#8211; f(a)) \/ h<\/li>\n\n\n\n<li>Substitute the expressions:\u00a0[ (1 \/ (a + h + 5)) &#8211; (1 \/ (a + 5)) ] \/ h<\/li>\n<\/ul>\n\n\n\n<p>This is a complex fraction. The first step to simplifying it is to combine the two fractions in the numerator. To do this, we find a common denominator, which is the product of the two individual denominators:&nbsp;(a + h + 5)(a + 5).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Rewrite the numerator with the common denominator:<br>[ (1 * (a + 5)) \/ ((a + h + 5)(a + 5)) ] &#8211; [ (1 * (a + h + 5)) \/ ((a + h + 5)(a + 5)) ]<\/li>\n\n\n\n<li>Combine the fractions over the single denominator:<br>( (a + 5) &#8211; (a + h + 5) ) \/ ( (a + h + 5)(a + 5) )<\/li>\n\n\n\n<li>Simplify the new numerator by distributing the negative sign and combining like terms:<br>( a + 5 &#8211; a &#8211; h &#8211; 5 ) \/ ( (a + h + 5)(a + 5) )<br>The\u00a0a\u00a0and\u00a0-a\u00a0cancel out, and the\u00a05\u00a0and\u00a0-5\u00a0cancel out, leaving just\u00a0-h.<\/li>\n\n\n\n<li>The simplified numerator is:\u00a0-h \/ ( (a + h + 5)(a + 5) )<\/li>\n<\/ul>\n\n\n\n<p>Now, we place this back into the full difference quotient expression:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( -h \/ ((a + h + 5)(a + 5)) ) \/ h<\/li>\n<\/ul>\n\n\n\n<p>Dividing by&nbsp;h&nbsp;is the same as multiplying by its reciprocal,&nbsp;1\/h.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( -h \/ ((a + h + 5)(a + 5)) ) * (1 \/ h)<\/li>\n<\/ul>\n\n\n\n<p>Finally, we can cancel the&nbsp;h&nbsp;in the numerator with the&nbsp;h&nbsp;in the denominator, which leaves&nbsp;-1&nbsp;in the numerator.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The final simplified answer is:\u00a0-1 \/ ( (a + h + 5)(a + 5) )<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-314.jpeg\" alt=\"\" class=\"wp-image-243263\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp;points) Given the function&nbsp;f(x)=1x+5, calculate the following values: The Correct Answer and Explanation is: Of course. Here are the correct answers and a detailed explanation for the problem. Answers: Explanation: This problem asks you to work with function notation and then to calculate and simplify the difference quotient, which is a fundamental concept in calculus [&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-243251","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243251","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=243251"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243251\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243251"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243251"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}