Answer
$f^{-1}(x)=\sqrt {r^2- x^2}$
Work Step by Step
In order to compute the inverse function, we must "interchange" $y$ and $x$ and then solve for the "new" $y$ (which is $f^{-1}(x)$).
Here, we have: $y=\sqrt {r^2-x^2} ; 0 \leq x \leq r$
Switch $x$ to $f^{-1} (x)$ and $y$ to $x$ in the function to obtain the inverse.
$x=\sqrt {r^2-y^2}$
$x^2=r^2-y^2\\ y^2= {r^2-x^2}$
Finally, solve for $y$ (which is $f^{-1}(x)$).
$f^{-1}(x)=\sqrt {r^2- x^2}$
