Answer
The graph is shown below.
Work Step by Step
$f\left( x \right)=-{{x}^{2}}$
Evaluate the inverse of the function $f\left( x \right)=-{{x}^{2}}$.
Replace the function $f\left( x \right)$ with y.
$y=-{{x}^{2}}$
Interchange the variables x and y.
$x=-{{y}^{2}}$
Solve for y value.
$\begin{align}
& {{y}^{2}}=-x \\
& y=\sqrt{-x} \\
\end{align}$
Replace y with ${{f}^{-1}}\left( x \right)$ as follows.
${{f}^{-1}}\left( x \right)=\sqrt{-x}$
Thus, the inverse function is ${{f}^{-1}}\left( x \right)=\sqrt{-x}$.
Since the function is $f\left( x \right)=-{{x}^{2}}$ where $x\ge 0$ the inverse is ${{f}^{-1}}\left( x \right)=\sqrt{-x}$ where $x\le 0$.
Consider the function.
$f\left( x \right)=-{{x}^{2}}$
Substitute $x=0,1,2$ in the function $f\left( x \right)=-{{x}^{2}}$.
For $x=0$, the value of $f\left( x \right)$ is,
$\begin{align}
& f\left( 0 \right)=-{{0}^{2}} \\
& =0
\end{align}$
For $x=1$, the value of $f\left( x \right)$ is,
$\begin{align}
& f\left( 1 \right)=-{{1}^{2}} \\
& =-1
\end{align}$
For $x=2$, the value of $f\left( x \right)$ is,
$\begin{align}
& f\left( 2 \right)=-{{2}^{2}} \\
& =-4
\end{align}$
Tabulate for the obtained values as shown below.
$\begin{matrix}
x & f\left( x \right) \\
0 & 0 \\
1 & -1 \\
2 & -4 \\
\end{matrix}$
Substitute $x=-9,-4,-1$ in the function ${{f}^{-1}}\left( x \right)=\sqrt{-x}$.
For $x=-9$, the value of $f\left( x \right)$ is,
$\begin{align}
& {{f}^{-1}}\left( -9 \right)=\sqrt{-\left( -9 \right)} \\
& =\sqrt{9} \\
& =3
\end{align}$
For $x=-4$, the value of $f\left( x \right)$ is,
$\begin{align}
& {{f}^{-1}}\left( -4 \right)=\sqrt{-\left( -4 \right)} \\
& =\sqrt{4} \\
& =2
\end{align}$
For $x=-1$, the value of $f\left( x \right)$ is,
$\begin{align}
& {{f}^{-1}}\left( -1 \right)=\sqrt{-\left( -1 \right)} \\
& =\sqrt{1} \\
& =1
\end{align}$
Tabulate for the obtained values as shown below.
$\begin{matrix}
x & f\left( x \right) \\
-9 & 3 \\
-4 & 2 \\
-1 & 1 \\
\end{matrix}$
Plot these points and sketch the graphs of the functions $f\left( x \right)=-{{x}^{2}}$ and ${{f}^{-1}}\left( x \right)=\sqrt{-x}$ as shown in the figure below.
