Solving functional equation $f(x+y)^2=f(x)^2+f(y)^2$

Question

I need to solve the following functional equation:$$f(x+y)^2=f(x)^2+f(y)^2$$ I'm familiar with simpler ones such as $f(x)+f\left(\frac{1}{1-x}\right)=x$ (I use substitutions), but here I cannot find any reasonable substitution.

Answer 1

Substitute $x=0$ to obtain $f(0)=0$. then, by substituting $y=-x$, we get $f(x)^2+f(-x)^2=0$ which implies $f(x)=0$ for all $x$.

Generally if you try to solve functional equations with two or more variables, try to start with substitutions for wich the expressions involved simplify to constants.