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 - Chapter Review - Review Exercises - Page 370: 13

Answer

$f^{-1}(x)=(x-1)^3$ $f(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$. $f^{-1}(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$.

Work Step by Step

1. $f(x)=x^{1/3}+1 \Longrightarrow y=x^{1/3}+1 \Longrightarrow x=y^{1/3}+1 \Longrightarrow y=(x-1)^3 \Longrightarrow f^{-1}(x)=(x-1)^3$ 2. check $f(f^{-1}(x))=((x-1)^3)^{1/3}+1=x$. $f^{-1}(f(x))=((x^{1/3}+1)-1)^3=x$. 3. The domain and the range of $f(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$. The domain and the range of $f^{-1}(x)$: $(-\infty,\infty)$ and $(-\infty,\infty)$.
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