How to find the range of functions like $f(x)=\sin (x) ^{sin(x)}$ on $(0,\Pi)$?
Usually, I find the inverse and then find the domain of the inverse function for the range of the original function, How do I find the inverse for this or is there any other way to find the range?
Suggestions: On $(0,\pi)$, $\sin(x)$ takes on all values in $(0,1]$. So your problem is equivalent to finding the max and min of $u^u$ for $u\in (0,1]$. By looking at derivatives, you should be able to do this using calculus techniques.
Hint: Our function is continuous. Evaluate at $\frac{\pi}{2}$ and $\pi$, and use the Intermediate Value Theorem. You will also need to show that the function takes no values outside the interval $[0,1]$.
Remark: The inverse function approach is not useful here. For one thing, our function is not one-to-one in our interval. Even if we restrict to an interval in which it is one-to-one, there will be no useful expression for the inverse function.
