What function satisfies $f(x)+f(−x)=f(x^2)$?

Question

What function satisfies $f(x)+f(−x)=f(x^2)$?

$f(x)=0$ is obviously a solution to the above functional equation.

We can assume f is continuous or differentiable or similar (if needed).

Answer 1

Define $f(x)$ any way you want for $x > 0$, then define $f(-x) = f(x^2) - f(x)$ also for $x>0$. If you want continuity, make sure that $\lim_{x\to 0} f(x) = 0$.

Answer 2

Give f(x) = ln(|x|) a try in your equation

Answer 3

I'm going to put my comment as an answer.

$\ln|1-x|$ seems to work: $$\ln|1-x|+\ln|1+x|=\ln|1-x^2|$$ Similarly, so does $\ln|1-x^3|$ (or any odd exponent).

Also, any linear combination of these works, as you can check. Thus: $$\ln(1+x+x^2)$$ works because it's equal to $\ln|1-x^3|-\ln|1-x|$. The example of $\ln(1+x+x^2)$ is nice because it's defined, continuous, and infinitely differentiable everywhere.

As far as I know, if you want it to be defined everywhere, continuous, and infinitely differentiable, this sort of thing is the only possible solution.