Group action defined on generators. In a question for an assignment (specific group but I've abstracted the question for academic integrity). I was given a group $G=\langle\ a,b \mid R\ \rangle$ where $R$ is some set of relations. Then, the question defines a group action on $X$ for each generating element. The first question is then "Verify that this gives a group action on $X$".
My problem is that it seems like by defining a group action on generators inherently assumes that $e(x)=x$ and $a(b(x))=ab(x)$ from the beginning, so it seems like there is nothing left to show? 
Since the question only defined the functions $a(x)=?$ and $b(c)=?$, I can't verify that $ab(x)=a(b(x))$, since I wasn't given the action defined by $ab$.
Is this a misunderstanding on my part, or do you think it is likely that the question itself is ambiguous?

The answer of ahulpke tells you how to show your map is a homomorphism in practice. I just wanted to tell you that you have seen this before.
In particular, one way to think about your question is that a group action of $G$ on a set $X$ corresponds to a homomorphism $G\rightarrow \operatorname{Sym}(X)$. You are, I am sure, well used to group homomorphisms being given only by the generators*. So the question you have is exactly the same game as showing a map is a homomorphism, just disguised slightly.
*e.g. "show that the map $\phi: \mathbb{Z}\rightarrow\mathbb{Z}, 1\mapsto -1$ is an isomorphism".
