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 - Page 258: 15


$\text{a) } f(g(x))=8x+29 \\\text{b) } g(f(x))=8x+16$

Work Step by Step

$\bf{\text{Solution Outline:}}$ With \begin{array}{l}\require{cancel} f(x)= 4x+5 \\g(x)= 2x+6 ,\end{array} to find $ f(g(x)) ,$ replace $x$ with $g(x)$ in $f.$ To find $g(f(x)),$ replace $x$ with $f(x)$ in $g.$ $\bf{\text{Solution Details:}}$ Replacing $x$ with $g(x)$ in $f$, then \begin{array}{l}\require{cancel} f(g(x))=f(2x+6) \\\\ f(g(x))=4(2x+6)+5 \\\\ f(g(x))=8x+24+5 \\\\ f(g(x))=8x+29 .\end{array} Replacing $x$ with $f(x)$ in $g$, then \begin{array}{l}\require{cancel} g(f(x))=g(4x+5) \\\\ g(f(x))=2(4x+5)+6 \\\\ g(f(x))=8x+10+6 \\\\ g(f(x))=8x+16 .\end{array} Hence, \begin{array}{l}\require{cancel} \text{a) } f(g(x))=8x+29 \\\text{b) } g(f(x))=8x+16 .\end{array}
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