Rationality testing and algebraic functions

Let S\subseteq\mathbb{C} be a finite set. A function f:\mathbb C -S\to\mathbb C is said to beĀ algebraicĀ if it satisfies a polynomial equation a_n(x)f(x)^n + \ldots + a_1(x) f(x) + a_0(x)=0 for a_i\in\mathbb{C}(x) rational functions.

Does there exist a funcion f_{\alpha,\beta}(x) such that f_{\alpha,\beta} is algebraic if and only \alpha and \beta are rational?

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Rationality testing and algebraic functions