Answer
$g^{-1}(x)=\sqrt[3]{27x}$
Work Step by Step
We are given the function:
$$g(x)=\dfrac{1}{27}x^3.$$
Find the inverse of the function:
$$\begin{align*}
g(x)&=\dfrac{1}{27}x^3\quad&&\text{Write original function.}\\
y&=\dfrac{1}{27}x^3\quad&&\text{Replace }f(x)\text{ by }y.\\
x&=\dfrac{1}{27}y^3\quad&&\text{Switch }x\text{ and }y.\\
27x&=y^3\quad&&\text{Multiply each side by }27.\\
\sqrt[3]{27x}&=y\quad&&\text{Take cubic root of each side.}
\end{align*}$$
The inverse of $g$ is $g^{-1}(x)=\sqrt[3]{27x}$.
Graph the function and its inverse:
