Solving this functional equation involving fixed polynomials

Question

Let $f, g \in \mathbb R[x]$ be fixed polynomials of the same degree $\deg f = \deg g = n$.

Can you solve the functional equation $(f\circ y)(x) = g(x-y(x))$ explicitly for $y: \mathbb R\to \mathbb R$?

Answer 1

Possible solution:

Let $f=g$ and let $y(x)=\frac{x}2$.

Then, $y(x)=x-y(x)$.