CatalinBordea wrote: Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations $$ \begin{matrix} f(g(x)+g(y))=f(g(x))+y \\ g(f(x)+f(y))=g(f(x))+y \end{matrix} , $$for all $ x,y\in\mathbb{Q} . $ 1) $f,g$ are bijective 2) $f(0)=g(0)=0$ 3) $f(g(x))=g(f(x))=x$ 4) $f,g$ are additive and so, since from $\mathbb Q\to\mathbb Q$, linear. 5) solutions are $\boxed{f(x)=ax\text{ and }g(x)=bx\quad\forall x}$ which indeed is a solution, $\forall a,b\in\mathbb Q$ such that $ab=1$

What’s y’ in this case?

It's a grammatical comma.

I have discussed this problem (with -y instead of y, but still the same idea) on my YouTube channel. Link: Video Solution