Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 5 - Exponential and Logarithmic Functions - 5.2 One-to-One Functions; Inverse Functions - 5.2 Assess Your Understanding - Page 268: 90

Answer

Domain: $[0,\infty)$, inverse:$f^{-1}(x)=\sqrt[4] x$.

Work Step by Step

We restrict the domain to $[0,\infty)$, (such that for every $y=f(x)$, there is a unique $x$, because we want it to pass the horizontal line test, because we can only find the inverse of a one-to-one function). If I want to find the inverse of a function, then I first must express $x$ from the function. Then, switching $x$ to $f^{−1}(x)$ and $y$ to $x$ in the function basically gives the inverse. Hence: $y=x^4\\\sqrt[4] y=x$ Therefore $f^{-1}(x)=\sqrt[4] x.$
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