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: 58

Answer

a) $\dfrac{amx+b}{cmx+d}$ b) $\dfrac{amx+bm}{cx+d}$ c) $D_{f\circ g}=\left(-\infty,-\dfrac{d}{cm}\right)\cup\left(-\dfrac{d}{cm},\infty\right)$ $D_{g\circ f}=\left(-\infty,-\dfrac{d}{c}\right)\cup\left(-\dfrac{d}{c},\infty\right)$ d) $m=1$

Work Step by Step

We are given the functions: $f(x)=\dfrac{ax+b}{cx+d}$ $g(x)=mx$ a) Determine $f\circ g$: $(f\circ g)(x)=f(g(x))=f(mx)=\dfrac{amx+b}{cmx+d}$ b) Determine $g\circ f$: $(g\circ f)(x)=g(f(x))=g\left(\dfrac{ax+b}{cx+d}\right)=m\cdot \dfrac{ax+b}{cx+d}=\dfrac{amx+bm}{cx+d}$ c) We have: $D_f=\left(-\infty,-\dfrac{d}{c}\right)\cup\left(-\dfrac{d}{c},\infty\right)$ $D_g=(-\infty,\infty)$ The domain of $f\circ g$ contains all the real values of $x$ except those for which $g(x)=-\dfrac{d}{c}$. $mx=-\dfrac{d}{c}$ $x=-\dfrac{d}{cm}$ $D_{f\circ g}=\left(-\infty,-\dfrac{d}{cm}\right)\cup\left(-\dfrac{d}{cm},\infty\right)$ The domain of $g\circ f$ contains all the real values of $x$ except $-\dfrac{d}{c}$. $D_{g\circ f}=\left(-\infty,-\dfrac{d}{c}\right)\cup\left(-\dfrac{d}{c},\infty\right)$ d) The domains of the two functions must be the same: $-\dfrac{d}{cm}=-\dfrac{d}{c}\Rightarrow m=1$ In order for $f\circ g=g\circ f$, we must have: $(f\circ g)(x)=(g\circ f)(x)$ for any real $x$ in the domain: $\dfrac{amx+b}{cmx+d}=\dfrac{amx+bm}{cx+d}$ But as $m=1$, the equality is true for all $x$ in the domain.
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