Answer
The graph is shown in below.
Work Step by Step
$g\left( x \right)=\frac{1}{2}{{x}^{3}}$
Evaluate the inverse of the function $g\left( x \right)=\frac{1}{2}{{x}^{3}}$.
Replace $g\left( x \right)$ with y.
$y=\frac{1}{2}{{x}^{3}}$
Interchange x and y.
$x=\frac{1}{2}{{y}^{3}}$
Solve for y value.
$\begin{align}
& 2x={{y}^{3}} \\
& y=\sqrt[3]{2x} \\
\end{align}$
Replace y with ${{g}^{-1}}\left( x \right)$ as follows.
${{g}^{-1}}\left( x \right)=\sqrt[3]{2x}$
Thus, the inverse function is ${{g}^{-1}}\left( x \right)=\sqrt[3]{2x}$.
Consider the function.
$g\left( x \right)=\frac{1}{2}{{x}^{3}}$
Substitute $x=-1.414,0,1.414$ in the function $g\left( x \right)=\frac{1}{2}{{x}^{3}}$.
For $x=-1.414$, the value of $g\left( x \right)$ is,
$\begin{align}
& g\left( -1.414 \right)=\frac{1}{2}{{\left( -1.414 \right)}^{3}} \\
& =-1.414
\end{align}$
For $x=0$, the value of $g\left( x \right)$ is,
$\begin{align}
& g\left( 0 \right)=\frac{1}{2}{{\left( 0 \right)}^{3}} \\
& =0
\end{align}$
For $x=1.414$, the value of $g\left( x \right)$ is,
$\begin{align}
& g\left( 1.414 \right)=\frac{1}{2}{{\left( 1.414 \right)}^{3}} \\
& =1.414
\end{align}$
Tabulate for obtained values as shown below.
$\begin{matrix}
x & g\left( x \right)=\frac{1}{2}{{x}^{3}} \\
-1.414 & -1.414 \\
0 & 0 \\
1.414 & 1.414 \\
\end{matrix}$
Consider the function.
${{g}^{-1}}\left( x \right)=\sqrt[3]{2x}$
Substitute $x=-1.414,0,1.414$ in the function ${{g}^{-1}}\left( x \right)=\sqrt[3]{2x}$.
For $x=-1.414$, the value of ${{g}^{-1}}\left( x \right)$ is,
$\begin{align}
& {{g}^{-1}}\left( -1.414 \right)=\sqrt[3]{2\left( -1.414 \right)} \\
& =\sqrt[3]{-2.828} \\
& =-1.414
\end{align}$
For $x=0$, the value of ${{g}^{-1}}\left( x \right)$ is,
$\begin{align}
& {{g}^{-1}}\left( 0 \right)=\sqrt[3]{2\left( 0 \right)} \\
& =\sqrt[3]{0} \\
& =0
\end{align}$
For $x=1.414$, the value of ${{g}^{-1}}\left( x \right)$ is,
$\begin{align}
& {{g}^{-1}}\left( 1.414 \right)=\sqrt[3]{2\left( 1.414 \right)} \\
& =\sqrt[3]{2.828} \\
& =1.414
\end{align}$
Tabulate for obtained values as shown below.
$\begin{matrix}
x & {{g}^{-1}}\left( x \right)=\sqrt[3]{2x} \\
-1.414 & -1.414 \\
0 & 0 \\
1.414 & 1.414 \\
\end{matrix}$
Plot these points and Sketch the graphs of the functions $g\left( x \right)=\frac{1}{2}{{x}^{3}}$ and ${{g}^{-1}}\left( x \right)=\sqrt[3]{2x}$ as shown in the figure below.
