Answer
$f(^{-1}(x)=x^3-7$
Work Step by Step
Let $y=f(x)$. Then the given one-to-one function, $
f(x)=\sqrt[3]{x+7}
,$ becomes
\begin{align*}\require{cancel}
y&=\sqrt[3]{x+7}
.\end{align*}
To find the inverse, interchange the $x$ and $y$ variables and then solve for $y$. That is,
\begin{align*}
x&=\sqrt[3]{y+7}
&(\text{interchange $x$ and $y$})
\\
(x)^3&=\left(\sqrt[3]{y+7}\right)^3
&(\text{solve for $y$})
\\
x^3&=y+7
\\
x^3-7&=y
.\end{align*}
Hence, the inverse of $
f(x)=\sqrt[3]{x+7}
$ is $
f(^{-1}(x)=x^3-7
$.
