Answer
$f^{-1}(x)=-\dfrac{5}{4}\sqrt[3]x$
Work Step by Step
We are given the function:
$$f(x)=-\dfrac{64}{125}x^3.$$
Find the inverse of the function:
$$\begin{align*}
f(x)&=-\dfrac{64}{125}x^3\quad&&\text{Write original function.}\\
y&=-\dfrac{64}{125}x^3\quad&&\text{Replace }f(x)\text{ by }y.\\
x&=-\dfrac{64}{125}y^3\quad&&\text{Switch }x\text{ and }y.\\
-\dfrac{125}{64}x&=y^3\quad&&\text{Multiply each side by }-\dfrac{125}{64}.\\
-\dfrac{5}{4}\sqrt[3]x&=y\quad&&\text{Take cubic root of each side.}
\end{align*}$$
The inverse of $f$ is $f^{-1}(x)=-\dfrac{5}{4}\sqrt[3]x$.
Graph the function and its inverse:
