In Wiki the definition of algebraic function is as following:
An algebraic function in degree $n$ is a function satisfy
$$ a_n(x)y^n + \cdots + a_{1}y + a_0 = 0.$$
My puzzle is that is there a restriction about what domain of an algebraic function should be? Since,we know that
$$ \dfrac 1x ,\ \sqrt{1-x^2},\ \dfrac{1}{\sqrt{1-x^3}} $$
etc.. Are all algebraic function, but with different domain.
Whether or not a function is algebraic is independent of its domain. There are algebraic functions whose domain is the whole space, i.e. $y=x$, and there are also algebraic functions with domain restrictions, i.e. $y=\frac{1}{x}$ for $x\neq 0$ or $y=\sqrt{1-x^2}$ for $x\in[-1,1]$.