Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.3 Exercises - Page 343: 4


$f(x)$ and $g(x)$ are inverse functions.

Work Step by Step

Let us find the inverse of $f(x)$ by solving for $x$. \[ y = 1-x^3 \\ \\ y-1 = -x^3 \\ \\ 1-y = x^3 \\ \\ \sqrt[3]{1-y} = x \] Therefore, the inverse of $f(x)$ is $\sqrt[3]{1-x}$, which is $g(x)$ so $f(x)$ is the inverse of $g(x)$ and vice versa. Graphically, we see that every point on $f(x)$, when reflected over the line $g(x)$ lies on one point on $g(x)$. Specifically, if a point $(a,b)$ is on $f(x)$, then there is a point $(b, a)$ on $g(x)$.
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