What's the solution of the functional equation $ f \big( g ( x ) + y \big) = g \big( f ( x ) + x \big) $?

Question

I need help with this:

Find functions $ f , g : \mathbb Z \to \mathbb Z $, knowing that $ g $ is injective and such that: $$ f \big( g ( x ) + y \big) = g \big( f ( x ) + x \big) \text{, for all } x, y \in \mathbb Z \text . $$

Answer 1

There must be an error on the question. As the second member doesn't depend on $y$ then $f$ must be a constant $c$.

$f(g(0) + y) = g(f(0)) = c, \mbox{ for all } y \in \mathbb{Z}$

Then the first member of the equation is a constant, $f(g(x) + y)=c$, so $g$ must also be constant, $c = g(c + x) \mbox{ for all } x \in \mathbb{Z}$.

Maybe you mean $f(g(x)+y) = g(f(y)+x), \mbox{ for all } x, y \in \mathbb{Z}$ ?