Answer
$f^{-1}(x)=\sqrt[3]{4-x}$
Work Step by Step
We are given the function:
$$f(x)=-x^3+4.$$
Find the inverse of the function:
$$\begin{align*}
f(x)&=-x^3+4\quad&&\text{Write original function.}\\
y&=-x^3+4\quad&&\text{Replace }f(x)\text{ by }y.\\
x&=-y^3+4\quad&&\text{Switch }x\text{ and }y.\\
x-4&=-y^3\quad&&\text{Subtract }4\text{ from beach side. }\\
4-x&=y^3\quad&&\text{Multiply each side by }-1.\\
\sqrt[3]{4-x}&=y\quad&&\text{Take cubic root of each side.}
\end{align*}$$
The inverse of $f$ is $f^{-1}(x)=\sqrt[3]{4-x}$.
Graph the function and its inverse:
