I am supposed to find period of function following the below relation if exists, for domain belonging to all real numbers.
$$f(x+T) = 1 + \left(2f(x)-f(x)^2\right)^{\frac12}$$
I just took $1$ to other side and squared and then differentiated to get $f(x+T) = f(-x)$ which indicates that period is $2T$.
Query: I don't think differentiation is a good way, because it is not mentioned that $f(x)$ is differentiable. Can someone please help in clarifying the underlying ideas or suggesting a altogether different approach.