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: 43

Answer

(a) $ x$, domain $\{x|x\ge1 \}$. (b) $ |x|$, domain $\{x|x=all\ real\ numbers \}$. (c) $ x^4+2x^2+2$, domain $\{x|x=all\ real\ numbers \}$. (d) $ \sqrt {\sqrt {x-1}-1}$, domain $\{x|x\ge2 \}$.

Work Step by Step

Given $f(x)=x^2+1$ and $g(x)=\sqrt {x-1}$, we have: (a) $f\circ g=(\sqrt {x-1})^2+1=x$, domain $\{x|x\ge1 \}$. (b) $g\circ f=\sqrt {(x^2+1)-1}=\sqrt {x^2}=|x|$, domain $\{x|x=all\ real\ numbers \}$. (c) $f\circ f=(x^2+1)^2+1=x^4+2x^2+2$, domain $\{x|x=all\ real\ numbers \}$. (d) $g\circ g=\sqrt {\sqrt {x-1}-1}$, domain $\{x|x\ge2 \}$.
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