Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 5 - Exponential and Logarithmic Functions - 5.1 Composite Functions - 5.1 Assess Your Understanding - Page 255: 57


a) $acx+ad+b$ b) $acx+bc+d$ c) $D_{f\circ g}=D_{g\circ f}=(-\infty,\infty)$ d) $d(a-1)=b(c-1)$

Work Step by Step

We are given the functions: $f(x)=ax+b$ $g(x)=cx+d$ a) Determine $f\circ g$: $(f\circ g)(x)=f(g(x))=f(cx+d)=a(cx+d)+b=acx+ad+b$ b) Determine $g\circ f$: $(g\circ f)(x)=g(f(x))=g(ax+b)=c(ax+b)+d=acx+bc+d$ c) As both functions are polynomials, the domain of $f\circ g$ and the domain of $g\circ f$ are the set of all real numbers: $D_{f\circ g}=D_{g\circ f}=(-\infty,\infty)$ d) The domains of the two functions are the same according to part c). In order for $f\circ g=g\circ f$, we must have: $(f\circ g)(x)=(g\circ f)(x)$ for any real $x$ $acx+ad+b=acx+bc+d$ Identify coefficients: $ad+b=bc+d$ $ad-d=bc-b$ $d(a-1)=b(c-1)$
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