(In reply to

heuristic solution by Charlie)

The solution found by Charlie (x=½(1+√13)) verifies f(x)=√4+√4-x=x, so the original problem asks for f(f(x))=x. Given this solution, the problem could also have asked for x such that f(f(f(x)))=x, or f(f(f(f(x))))=x, and so on.

A point: if f(x)=x, then obviously f(f(x))=x, but you should prove that no other x such that f(x)≠x can solve the original problem.