College Algebra, Chapter 3, 3.7, Section 3.7, Problem 50

Find the inverse of $\displaystyle f(x) = (2 - x^3)^5$

To find the inverse of $f(x)$, we write $y = f(x)$


$
\begin{equation}
\begin{aligned}

y =& (2 - x^3)^5
&& \text{Solve for $x$, take the fifth root of both sides}
\\
\\
\sqrt[5]{y} =& 2 - x^3
&& \text{Add $x^3$ and subtract } \sqrt[5]{y}
\\
\\
x^3 =& 2 - \sqrt[5]{y}
&& \text{Take the cube root}
\\
\\
x =& \sqrt[3]{2 - \sqrt[5]{y}}
&& \text{Interchange $x$ and $y$}
\\
\\
y =& \sqrt[3]{2 - \sqrt[5]{x}}
&&

\end{aligned}
\end{equation}
$


Thus, the inverse of $f(x) = (2 - x^3)^5$ is $\displaystyle f^{-1} (x) = \sqrt[3]{2 - \sqrt[5]{x}}$.