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: 39

Answer

$\text{a) } f(g(2))=21 \\\\\text{b) } g(f(2))=69$

Work Step by Step

$\bf{\text{Solution Outline:}}$ With \begin{array}{l}\require{cancel} f(x)= x+2 \\g(x)= 4x^2+x+1 ,\end{array} to find $ f(g(2)) ,$ find first $ g(2) .$ Then substitute the result in $f.$ To find $ g(f(2)) ,$ find first $ f(2) .$ Then substitute the result in $g.$ $\bf{\text{Solution Details:}}$ a) Replacing $x$ with $ 2 $ in $g$ results to \begin{array}{l}\require{cancel} g(x)=4x^2+x+1 \\\\ g(2)=4(2)^2+2+1 \\\\ g(2)=4(4)+3 \\\\ g(2)=16+3 \\\\ g(2)=19 .\end{array} Replacing $x$ with the result above in $f$ results to \begin{array}{l}\require{cancel} f(x)=x+2 \\\\ f(19)=19+2 \\\\ f(19)=21 .\end{array} Hence, $ f(g(2))=21 .$ b) Replacing $x$ with $ 2 $ in $f$ results to \begin{array}{l}\require{cancel} f(x)=x+2 \\\\ f(2)=2+2 \\\\ f(2)=4 .\end{array} Replacing $x$ with the result above in $g$ results to \begin{array}{l}\require{cancel} g(x)=4x^2+x+1 \\\\ g(4)=4(4)^2+4+1 \\\\ g(4)=4(16)+5 \\\\ g(4)=64+5 \\\\ g(4)=69 .\end{array} Hence, $ g(f(2))=69 .$ Therefore, \begin{array}{l}\require{cancel} \text{a) } f(g(2))=21 \\\\\text{b) } g(f(2))=69 .\end{array}
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