Answer
${{f}^{-1}}\left( x \right)=\frac{\sqrt[3]{x}}{3}$.
Work Step by Step
$f\left( x \right)=27{{x}^{3}}$
Evaluate the inverse of the function $f\left( x \right)=27{{x}^{3}}$ as follows.
Replace the function $f\left( x \right)$ with y.
$y=27{{x}^{3}}$
Interchange the variables x and y.
$x=27{{y}^{3}}$
Solve for the value of y.
$\begin{align}
& x=27{{y}^{3}} \\
& {{y}^{3}}=\frac{x}{27} \\
& y=\sqrt[3]{\frac{x}{27}} \\
& y=\frac{\sqrt[3]{x}}{3}
\end{align}$
Replace y with ${{f}^{-1}}\left( x \right)$ as follows.
${{f}^{-1}}\left( x \right)=\frac{\sqrt[3]{x}}{3}$
