# Funcional Equations:I'm confused [duplicate]

This question already has an answer here:

I need help with this:

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

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## marked as duplicate by Ross Millikan, rschwieb, Arkamis, Asaf Karagila, Stefan HansenFeb 14 '13 at 6:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

I note that this is not quite a duplicate, as the variables are different. It may be that the techniques of the cited answer apply as well. Certainly $f(x)=g(x)=x$ is again a solution. –  Ross Millikan Feb 13 '13 at 23:09
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## 1 Answer

For the first, note that the right side is independent of $y$, so $f(x)$ is constant. Then we have $f(0)=g(f(0)+x)$ with the left side independent of $x$. Then $g$ must be constant, but it must be injective. No solution.

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