parmenides51 wrote: Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$ and such that $g(x) = g(y)$ only if $x = y$. Let $P(x,y)$ be the assertion $f(g(x)+y)=g(f(y)+x)$ Subtracting $P(x,g(0))$ from $P(0,g(x))$ and using injectivity of $g(x)$, we get : $f(g(x))=x+f(g(0))$ But $P(x,0)$ $\implies$ $f(g(x))=g(x+f(0))$ And so $g(x+f(0))=x+f(g(0))$ And so $g(x)=x+a$ for some $a=f(g(0))-f(0)$ So $f(x+a)=f(g(x))=x+f(g(0))$ and $f(x)=x+b$ for some $b=f(g(0))-a$ Plugging this back in original equation, we get that any $a,b$ fits and so the solution $\boxed{f(x)=x+b\text{ and }g(x)=x+a\quad\forall x}$ whatever are $a,b\in\mathbb Z$