Find functions $f,g$ such that $f(x+y)=g(x·y)$ for all $x,y$?

Question

My approach towards this question was that first I putted x=0 and then y=0 which yields f(x)=f(y)and f(x)=g(0), and again for x=1 & y=1 it gives g (x)=f(1+x), and so on. My query is that what should the compact & rigorous form of proof or solution looks like because I found myself stuck between results and not comprehending results very well. Thanks.

Answer 1

$$f(x) = f(x + 0) = g(x \cdot 0) = g(0)$$ for all $x$. Thus $f(x)$ is constant and always equal to $g(0)$. Moreover, $$g(0) = f(1 + y) = g(1 \cdot y) = g(y)$$ for all $y$. Thus $g$ is also constant and equal to $g(0)$. Thus $f, g$ are the same constant function. All constant functions $f = g = c$ will satisfy the relation. This is all you need.