Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 3 - Exponents, Polynomials and Functions - 3.3 Composing Functions - 3.3 Exercises: 40

Answer

$\text{a) } f(g(4))=48 \\\\\text{b) } g(f(4))=528$

Work Step by Step

$\bf{\text{Solution Outline:}}$ With \begin{array}{l}\require{cancel} f(x)= 5x-2 \\g(x)= 2x^2-7x+6 ,\end{array} to find $ f(g(4)) ,$ find first $ g(4) .$ Then substitute the result in $f.$ To find $ g(f(4)) ,$ find first $ f(4) .$ Then substitute the result in $g.$ $\bf{\text{Solution Details:}}$ a) Replacing $x$ with $ 4 $ in $g$ results to \begin{array}{l}\require{cancel} g(x)=2x^2-7x+6 \\\\ g(4)=2(4)^2-7(4)+6 \\\\ g(4)=2(16)-28+6 \\\\ g(4)=32-28+6 \\\\ g(4)=10 .\end{array} Replacing $x$ with the result above in $f$ results to \begin{array}{l}\require{cancel} f(x)=5x-2 \\\\ f(10)=5(10)-2 \\\\ f(10)=50-2 \\\\ f(10)=48 .\end{array} Hence, $ f(g(4))=48 .$ b) Replacing $x$ with $ 4 $ in $f$ results to \begin{array}{l}\require{cancel} f(x)=5x-2 \\\\ f(4)=5(4)-2 \\\\ f(4)=20-2 \\\\ f(4)=18 .\end{array} Replacing $x$ with the result above in $g$ results to \begin{array}{l}\require{cancel} g(x)=2x^2-7x+6 \\\\ g(18)=2(18)^2-7(18)+6 \\\\ g(18)=2(324)-126+6 \\\\ g(18)=648-126+6 \\\\ g(18)=528 .\end{array} Hence, $ g(f(4))=528 .$ Therefore, \begin{array}{l}\require{cancel} \text{a) } f(g(4))=48 \\\\\text{b) } g(f(4))=528 .\end{array}
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