Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 4 - Exponential and Logarithmic Functions - Section 4.1 Composite Functions - 4.1 Assess Your Understanding - Page 280: 39

Answer

(a) $ \frac{4}{x+4}$, domain $\{x|x\ne-4,0 \}$. (b) $ \frac{-4x+4}{x}$, domain $\{x|x\ne0,1 \}$. (c) $ x$, domain $\{x|x\ne1 \}$. (d) $ x$, domain $\{x|x\ne0 \}$.

Work Step by Step

Given $f(x)=\frac{x}{x-1}$ and $g(x)=-\frac{4}{x}$, we have: (a) $f\circ g=\frac{-\frac{4}{x}}{-\frac{4}{x}-1}=\frac{4}{x+4}$, domain $\{x|x\ne-4,0 \}$. (b) $g\circ f=-\frac{4}{\frac{x}{x-1}}=\frac{-4x+4}{x}$, domain $\{x|x\ne0,1 \}$. (c) $f\circ f=\frac{\frac{x}{x-1}}{\frac{x}{x-1}-1}=x$, domain $\{x|x\ne1 \}$. (d) $g\circ g=-\frac{4}{-\frac{4}{x}}=x$, domain $\{x|x\ne0 \}$.
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