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How do you know that the domain is the set of integers?
If f is multiplicative, then $\displaystyle f(k)f(n) = f(kn)$ for all k,n. Then $\displaystyle f(n) = \frac{f(kn)}{f(k)}$, which is also multiplicative. Also we know that $\displaystyle f(k) \neq 0$ so we don't have to worry about dividing by zero or anything like that...