Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 2 - 2.6 - Combinations of Functions: Composite Functions - 2.6 Exercises - Page 219: 36


a) $\sqrt[3]{x^3-4}$; domain $(-\infty,\infty)$ b) $x-4$; domain: $(-\infty,\infty)$

Work Step by Step

We are given the functions: $f(x)=\sqrt[3]{x-5}$ $g(x)=x^3+1$ Determine the domains $D_f$ and $D_g$ of the two functions: $D_f=(-\infty,\infty)$ $D_g=(-\infty,\infty)$ a) Find $f\circ g$ and its domain $D_{f\circ g}$: $(f\circ g)(x)=f(g(x))=f\left(x^3+1\right)=\sqrt[3]{x^3+1-5}=\sqrt[3]{x^3-4}$ $D_{f\circ g}=(-\infty,\infty)$ b) Find $g\circ f$ and its domain $D_{g\circ f}$: $(g\circ f)(x)=g(f(x))=g(\sqrt[3]{x-5})=(\sqrt[3]{x-5})^3+1=x-5+1=x-4$ $D_{g\circ f}=(-\infty,\infty)$
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