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).
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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.