Answer
The graph is shown below.
Work Step by Step
$f\left( x \right)={{x}^{2}}-1$
Evaluate the inverse of the function $f\left( x \right)={{x}^{2}}-1$ as follows.
Replace the function $f\left( x \right)$ with y.
$y={{x}^{2}}-1$
Interchange the variables x and y.
$x={{y}^{2}}-1$
Solve for y value when $x\le 0$.
$\begin{align}
& {{y}^{2}}=x+1 \\
& y=-\sqrt{x+1} \\
\end{align}$
Replace y with ${{f}^{-1}}\left( x \right)$ as follows.
${{f}^{-1}}\left( x \right)=-\sqrt{x+1}$
Thus, the inverse function is ${{f}^{-1}}\left( x \right)=-\sqrt{x+1}$.
Sketch the graphs of the functions $f\left( x \right)={{x}^{2}}-1$ and ${{f}^{-1}}\left( x \right)=-\sqrt{x+1}$ as shown in the figure below.
