Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - Chapter Review Exercises - Page 37: 40


$$\cos \left(x^{-1}\right)$$ Domain: $\{x: x \neq 0\}$ $$ \ (\cos x)^{-1} $$ Domain: $\{x: x \neq (2 k+1) \frac{\pi}{2}\} $

Work Step by Step

We are given the functions: $$ h(x)=\cos x \text { and } g(x)=x^{-1}$$ We find the composite function as: \begin{align*} (h \circ g)(x)&=h(g(x))\\ &=h\left(x^{-1}\right)\\ &=\cos \left(x^{-1}\right) \end{align*} The domain is $\{x: x \neq 0\}$ Next, we find: \begin{align*} (g \circ h)(x)&=g(h(x))\\ &=g(\cos x)\\ &=(\cos x)^{-1} \end{align*} Domain is $\{x: \cos x \neq 0\}$, so $\{x: x \neq (2 k+1) \frac{\pi}{2}\} $
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