Evaluate $\tan^{-1} (\tan (x + \frac{\pi}{3}))$, where $x\in (0,\frac{\pi}{2})$

Question

This is a simplified version of a much complex question, but the problem I have is mainly conceptual.

Going by the form we have, the answer should simply be $x+\frac{\pi}{3}$, but it doesn’t account for the fact that when $x\to \frac{\pi}{2}$, the domain of the function isn’t satisfied

Do we need to make any changes to angle to fit it into the proper domain, or is writing it directly fine?

Edit: According to the answer, it is supposed to be written directly

Answer 1

For me, it should be written as \begin{cases} x+\dfrac \pi 3&\text{ if}\quad 0<x<\frac\pi6 ,\\[1ex] x-\dfrac{2\pi}3 &\text{ if}\quad \frac\pi6 <x <\frac\pi 2. \end{cases}