Consider a function $g: \mathbb{R}^2 \to \mathbb{R}$ and another function $h: \mathbb{R}^2 \to \mathbb{R}$. Does there always exist a function $f$ s.t. $g(f(x),f(y)) = h(x,y)$ ?
I can construct simple cases eg. $g(x,y)=x+y$, $h(x,y)=x^2+y^2+xy$ and then put values to get contradiction in $f(x) + f(y) = x^2 + y^2 + xy$ but I am not sure what is the correct approach to solve the general problem.
I see that for $g: \mathbb{R} \to \mathbb{R}$ this is straightforward as $f(x) = g^{-1}h(x)$, so $f$ exists if $g$ is invertible.
To prove something is not true it is enough to construct an example for which it fails.
Let $g(x,y) = x$ and $h(x,y)=y$. Can you get an $f$ to satisfy your condition?