I would be inclined to look at the basic "sets of numbers" and use their defining properties. For example, the "natural numbers" or "counting numbers" are defined by "induction"- 1 is a natural number and, if n is a natural number, then n+ 1 is also. Obviously, for any number, x, f(x)= f(x)(1). Use induction to show that for any positive integer, n, then f(n)= f(x)n and then take x= 1. (You will use the general "x" below.)
The "whole numbers" are just the positive integers together with 0. And 0 is the "additive identity", for any positive integer, n, n+ 0= n. Look at f(n+ 0)= f(n)+ f(0).
The "integers" includes the negative integers: we now have additive inverse. Any negative integer is of the form -n for some positive integer n. f(n+ (-n))= f(n)+ f(-n) and, of course, f(n+(-n))= f(0).
Finally, look at numbers of the form 1/n, n non-zero. Obviously, n(1/n)= 1 so f(n(1/n))= f(1). But you have already shown that f(nx)= f(x)n. So take x= 1/n.