There is a theorem that says rational functions in the extended complex plane are exactly the meromorphic functions.
After this, my textbook draws the corollary: "...as a consequence, a rational function is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles."
I don't see how it follows. Plus I think I have an example. Take a rational function with no zeros and two poles of multiplicity one, at 0 and 1.
At least two such functions exist:
$f(x)=\frac{1}{x} + \frac{1}{x-1}$
$g(x)=\frac{1}{x} + \frac{2}{x-1}$