You can't do this for general functions, but it's fine to do in cases where the definition of f naturally embeds into the rationals. For example, a polynomial over Z is also a polynomial over Q or C.
The movement from R to C can be done rigorously. It gets hand-waved away in more application-oriented math courses, but it's done properly in higher level theoretically-focused courses. Lifting from a smaller field (or other algebraic structure) to a larger one is a very powerful idea because it often reveals more structure that is not visible in the smaller field. Some good examples are using complex eigenvalues to understand real matrices, or using complex analysis to evaluate integrals over R.
You can go farther and say that you can't even construct real numbers without strong enough axioms. Theories of first order arithmetic, like Peano arithmetic, can talk about computable reals but not reals in general.
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