Nonlinear Functional Equation $f(x)^{2}=f\left(\frac x a\right)$

Question

Find all functions $f(x)$ such for a given fixed $a\in \mathbb{R}$ such that the following functional equation holds: $$f(x)^{2}=f\left(\frac x a\right)$$

I'm not sure how to solve this equation other then using the method of power series, any tips?

Answer 1

I will describe 2 trivial functions and one nontrivial family of functions satisfying the above relation for each $a$. I will give this answer even though I am not finding all such functions because nobody else has provided an answer in a day.

First let $c = \frac{1}{a}$. I think it's more natural to think about the equation as $f(x)^2 = f(cx)$ than as in the OP. Clearly $f(x) = 0$ and $f(x) = 1$ are solutions.

More interestingly, the family of functions $$f(x) = p^{(qx)^{\log_c(2)}}$$ for any $p, q > 0$ satisfies the inequality, which we can see as follows:

$$f(x)^2 = p^{2\cdot (qx)^{log_c(2)}} = 2^{c^{\log_c(2)} (qx)^{\log_c(2)}} = p^{(q(cx))^{\log_c(2)}} = f(cx).$$

This approach immediately generalizes to solve equations of the form $$f(x)^n = f(cx),$$ with solution $$f(x) = p^{(qx)^{\log_c(n)}}.$$