#### Answer

$f(x)$ has an inverse function on the given interval.

#### Work Step by Step

We find the derivative of $f(x)$:
\[ f'(x) = - \csc ^2 (x) = - \frac{1}{\sin^2(x)} \]
Notice that on the domain $(0, \pi)$, $\sin(x)$ is always positive. Therefore, on the same domain, $f'(x)$ is always negative, and $f(x)$ is strictly monotonic.