Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 2 - 2.6 - Combinations of Functions: Composite Functions - 2.6 Exercises - Page 221: 71


a) $f(x)=2x+3, g(x)=3x+6$ b) $f(x)=x^5, g(x)=x^2$

Work Step by Step

a) Let's note: $f(x)=ax+b$ $g(x)=cx+d$ Determine $f\circ g$ and $g\circ f$: $(f\circ g)(x)=f(g(x))=f(cx+d)=a(cx+d)+b=acx+ad+b$ $(g\circ f)(x)=g(f(x))=g(ax+b)=c(ax+b)+d=acx+bc+d$ In order to have $f\circ g=g\circ f$, we must have: $acx+ad+b=acx+bc+d$ $ad+b=bc+d$ $ad-d=bc-b$ $d(a-1)=b(c-1)$ For example: $a=2$ $d=6$ $c=3$ $b=3$ $f(x)=2x+3$ $g(x)=3x+6$ b) Let's have: $f(x)=x^5$ $g(x)=x^2$ Determine $f\circ g$ and $g\circ f$: $(f\circ g)(x)=f(g(x))=f(x^2)=(x^2)^5=x^{10}$ $(g\circ f)(x)=g(f(x))=g(x^5)=(x^5)^2=x^{10}$
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