Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Section 1.5 - Inverse Functions and Logarithms - 1.5 Exercises - Page 67: 32

Answer

(a) The inverse function is $~~g^{-1}(x) = \sqrt[3] {1-x^3}$ $g^{-1}(x) = g(x)$ (b) If we reflect the graph of $g(x)$ in the line $y=x$, we will get exactly the same graph. The function $g(x)$ is its own inverse.
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Work Step by Step

(a) We can solve $g$ for $x$: $y = \sqrt[3] {1-x^3}$ $y^3 = 1-x^3$ $y^3-1 = -x^3$ $x^3 = 1-y^3$ $x = \sqrt[3] {1-y^3}$ We reverse the places of $x$ and $y$: $y = \sqrt[3] {1-x^3}$ The inverse function is $~~g^{-1}(x) = \sqrt[3] {1-x^3}$ $g^{-1}(x) = g(x)$ (b) If we reflect the graph of $g(x)$ in the line $y=x$, we will get exactly the same graph. The function $g(x)$ is its own inverse.
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