Chapter 5 - Exponential and Logarithmic Functions - 5.2 One-to-One Functions; Inverse Functions - 5.2 Assess Your Understanding - Page 268: 86

$f^{-1}(x)=\sqrt{r^2-x^2}$

If I want to find the inverse of a function, then I first must express $x$ from the function. Then, switching $x$ to $f^{-1}(x)$ and $y$ to $x$ in the function basically gives the inverse. $y=\sqrt{r^2-x^2}$, $0\leq x \leq r$. Thus: $y^2=r^2-x^2\\x^2=r^2-y^2\\x=\sqrt{r^2-y^2}$ Hence $f^{-1}(x)=\sqrt{r^2-x^2}$