University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.6 - Inverse Functions and Logarithms - Exercises - Page 49: 24



Work Step by Step

$$y=f(x)=x^{2/3}\hspace{1cm}x\ge0$$ To find its inverse: 1) Solve for $x$ in terms of $y$: $$y=x^{2/3}$$ $$y = \sqrt[3]{x^2}$$ $$x^2=y^3$$ Since $x\ge0$, we take here only the positive values of $x$, which means $$x=\sqrt{y^3}=y^{3/2}$$ 2) Interchange $x$ and $y$: $$y=x^{3/2}$$ Therefore, the inverse of function $y=f(x)=x^{2/3}$, $x\ge0$ is the function $y=x^{3/2}$. In other words, $$f^{-1}(x)=x^{3/2}$$
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