Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 386: 4


The inverse function is $g(x)=(x^2+8)^{1/3}$; its domain is $[0,\infty)$ and range is $(2,\infty)$.

Work Step by Step

To find the inverse of $y=\sqrt{x^3-8}$, we have $$y^2=x^3-8 \Longrightarrow x^3=y^2+8\Longrightarrow x=(y^2+8)^{1/3}.$$ Hence, the inverse function is $g(x)=(x^2+8)^{1/3}$. Note that the lowest value that $x$ can take on is $0$, which gives the range of $(2,\infty)$. The domain is the range of the original function, which is $[0,\infty)$.
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