I came across the following problem and I have been cracking my head with it:
Prove or disprove that there are no functions $f$ and $g$ such that
$$f(x,g(y-x)+g(z-x))=y^2+z^2$$
for all $x,y,z \in \mathbb{R}$.
The point is that I never had contact with functional equations before so I don't even have the idea of how to start this.
However I did try some things, but I don't really know what to do with them exactly because I lack theory.
Considering $y=z=\alpha$, leaving $x$ free, this yields:
$$f(x,2g(\alpha-x))=2\alpha^2$$
Also, if I consider $x=y=\beta$, leaving $z$ free, this yields:
$$f(\beta,g(0)+g(z-\beta))=\beta^2+z^2$$
And, if I consider $x=z=\gamma$, leaving $y$ free, this yields:
$$f(\gamma,g(y-\gamma)+g(0))=y^2+\gamma^2$$
Finally, if I consider $x=y=z=\delta$:
$$f(\delta,2g(0))=2\delta^2$$
Could you help? Also, would appreciate some suggestions for basic literature in functional equations.