Functional equation $f(x^2+y)=f(x)+f(y^2)$ from Olympiad

Question

Let $f: \mathbb{R} \to \mathbb{R}$ $$f(x^2+y)=f(x)+f(y^2)$$

How do I find all functions that fulfill this equation?

I tried to just write many equalities but it just doesnt help.

Answer 1

You can solve this pretty immediately by looking at the cases $x=0$, $y=0$, and $x^2+y=0$. (The only possibility is the constant zero function.)

Answer 2

I would try to see how it behaves at "special points" like +-1, 0 and some others, and see how it can behave.

For example, if you let $y=0$ this becomes $f(x^2) = f(x)$.

Then for $x=y=1$ you get $f(2) = 2f(1)$.

For $x=1, y=-1$ you get $f(0) = 2f(1)$ also...

Try a few more combinations and you'll get enough constraints to define $f$, or reach a contradiction.