Answer
The graph is shown in below.
Work Step by Step
$g\left( x \right)=\frac{1}{3}{{x}^{3}}$
Evaluate the inverse of the function $g\left( x \right)=\frac{1}{3}{{x}^{3}}$ as follows.
Replace the function $g\left( x \right)$ with y.
$y=\frac{1}{3}{{x}^{3}}$
Interchange the variables x and y.
$x=\frac{1}{3}{{y}^{3}}$
Solve for y value.
$\begin{align}
& 3x={{y}^{3}} \\
& y=\sqrt[3]{3x} \\
\end{align}$
Replace y with ${{g}^{-1}}\left( x \right)$ as follows.
${{g}^{-1}}\left( x \right)=\sqrt[3]{3x}$
Thus, the inverse function is ${{g}^{-1}}\left( x \right)=\sqrt[3]{3x}$.
Consider the function.
$g\left( x \right)=\frac{1}{3}{{x}^{3}}$
Substitute $x=-1.732,0,1.732$ in the function $g\left( x \right)=\frac{1}{3}{{x}^{3}}$.
For $x=-1.732$, the value of $g\left( x \right)$ is,
$\begin{align}
& g\left( -1.732 \right)=\frac{1}{3}{{\left( -1.732 \right)}^{3}} \\
& =-1.732
\end{align}$
For $x=0$, the value of $g\left( x \right)$ is,
$\begin{align}
& g\left( 0 \right)=\frac{1}{3}{{\left( 0 \right)}^{3}} \\
& =0
\end{align}$
For $x=1.732$, the value of $g\left( x \right)$ is,
$\begin{align}
& g\left( 1.732 \right)=\frac{1}{3}{{\left( 1.732 \right)}^{3}} \\
& =1.732
\end{align}$
Tabulate for obtained values as shown below.
$\begin{matrix}
x & g\left( x \right)=\frac{1}{3}{{x}^{3}} \\
-1.732 & -1.732 \\
0 & 0 \\
1.732 & 1.732 \\
\end{matrix}$
Consider the function.
${{g}^{-1}}\left( x \right)=\sqrt[3]{3x}$
Substitute $x=-1.732,0,1.732$ in the function ${{g}^{-1}}\left( x \right)=\sqrt[3]{3x}$.
For $x=-1.732$, the value of ${{g}^{-1}}\left( x \right)$ is,
$\begin{align}
& {{g}^{-1}}\left( -1.732 \right)=\sqrt[3]{3\left( -1.732 \right)} \\
& =\sqrt[3]{-5.196} \\
& =-1.732
\end{align}$
For $x=0$, the value of ${{g}^{-1}}\left( x \right)$ is,
$\begin{align}
& {{g}^{-1}}\left( 0 \right)=\sqrt[3]{3\left( 0 \right)} \\
& =\sqrt[3]{0} \\
& =0
\end{align}$
For $x=1.732$, the value of ${{g}^{-1}}\left( x \right)$ is,
$\begin{align}
& {{g}^{-1}}\left( 0 \right)=\sqrt[3]{3\left( 1.732 \right)} \\
& =\sqrt[3]{5.196} \\
& =1.732
\end{align}$
Tabulate for obtained values as shown below.
$\begin{matrix}
x & {{g}^{-1}}\left( x \right)=\sqrt[3]{2x} \\
-1.732 & -1.732 \\
0 & 0 \\
1.732 & 1.732 \\
\end{matrix}$
Plot these points and sketch the graphs of the functions $g\left( x \right)=\frac{1}{3}{{x}^{3}}$ and ${{g}^{-1}}\left( x \right)=\sqrt[3]{3x}$ as shown in the figure below.
