Answer
The inverse function for $f\left( x \right)=1-{{x}^{2}},x\ge 0$ is ${{f}^{-1}}\left( x \right)=\sqrt{1-x}$ .
Work Step by Step
Consider the provided function:
$f\left( x \right)=1-{{x}^{2}},x\ge 0$
Let $y=1-{{x}^{2}}$
The steps to find the inverse function $y=f\left( x \right)$ are as follows:
Step 1: Interchange $x$ and $y$.
$x=1-{{y}^{2}}$
Step 2: Solve the equation for $y$ .
$\begin{align}
& x=1-{{y}^{2}} \\
& {{y}^{2}}=1-x \\
& y=\sqrt{1-x}
\end{align}$
Step 3: Replace $y$ with ${{f}^{-1}}\left( x \right)$ .
${{f}^{-1}}\left( x \right)=\sqrt{1-x}$
Therefore, the inverse function for $f\left( x \right)=1-{{x}^{2}},x\ge 0$ is ${{f}^{-1}}\left( x \right)=\sqrt{1-x}$.
