Hello !
Originally Posted by
magentarita Find the inverse of the function f(x) = sqrt{r^2 - x^2}, where 0 < or = to x < or = to r
Why are you stuck ? As $\displaystyle f$ is a bijection on $\displaystyle [0,r]$ (Can you show this ?) its inverse function exists. Let $\displaystyle y=f(x)$ : we're asked to find the inverse function of $\displaystyle f$ that is the function $\displaystyle g$ such that $\displaystyle x=g(y)$. This can simply be done by solving $\displaystyle y=\sqrt{r^2-x^2}$ for $\displaystyle x$ :
$\displaystyle \begin{aligned}
y=\sqrt{r^2-x^2} & \Longleftrightarrow y^2=r^2-x^2\\
&\Longleftrightarrow x^2=\ldots\\
&\Longleftrightarrow x=\ldots
\end{aligned}$
hence the inverse function of $\displaystyle f$ is ...