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: 24


$\text{a) } f(g(x))=4x^2-12x+41 \\\\\text{b) } g(f(x))=16x^2-68x+82$

Work Step by Step

$\bf{\text{Solution Outline:}}$ With \begin{array}{l}\require{cancel} f(x)= 4x-7 \\g(x)= x^2-3x+12 ,\end{array} replace $x$ with $g(x)$ in $f$ to find $f(g(x)).$ 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(x^2-3x+12) \\\\ f(g(x))=4(x^2-3x+12)-7 \\\\ f(g(x))=4x^2-12x+48-7 \\\\ f(g(x))=4x^2-12x+41 .\end{array} Replacing $x$ with $f(x)$ in $g.$ Hence, \begin{array}{l}\require{cancel} g(f(x))=g(4x-7) \\\\ g(f(x))=(4x-7)^2-3(4x-7)+12 \\\\ g(f(x))=(16x^2-56x+49)+(-12x+21)+12 \\\\ g(f(x))=16x^2-68x+82 .\end{array} Hence, \begin{array}{l}\require{cancel} \text{a) } f(g(x))=4x^2-12x+41 \\\\\text{b) } g(f(x))=16x^2-68x+82 .\end{array}
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