How to computer $f(\frac{1}{2})$ given $f(f(x)) = x^2 + \frac{1}{4}$?

Question

I have observed that $f(f(\frac{1}{2})) = \frac{1}{2}$ and $f(f(f(x))) = f(x^2 + \frac{1}{4})$, and when $x = \frac{1}{2}$, we have $f(\frac{1}{2}^2 + \frac{1}{4}) = f(\frac{1}{2})$. But I don't know how to proceed, or if any of these observations are helpful. How can I from this compute $f(\frac{1}{2})$ and what is the intuition or reasoning that would guide you towards the answer?

Answer 1

Denote $ c = f\left(\frac{1}{2}\right). $

We have $$ f\left(f\left( \frac{1}{2}\right)\right) = f(c) = \frac{1}{2}. $$ $$ f\left(f\left( c\right)\right) = f\left(\frac{1}{2}\right) = c^2 + \frac{1}{4} = c. $$ So $$ \left(c - \frac{1}{2} \right)^2 = 0. $$ This means that the only possible value for $c$ (if such function exists, of course) is $\frac{1}{2}$.