Why can't you have multiple domains in one function?

Question

Let's say we have function $y=\sqrt x$. For natural numbers it has two solutions. For example $\sqrt 4 = \pm2$. Wouldn't it make sense then to graph a sideways parabola with more than one points in vertical lines? Why mathematics say you can't do this when it is very obvious the square root of $x$ is only reflection of $x^2$ along the $y=x$ line?

Answer 1

This is called a multi-valued function. The principal branch, or the positive part of $\sqrt{x}$ is usually taken to be its definition.

A multi-valued function does not satisfy the criterion of many-to-one(every element in its domain is associated to exactly one element in its range), so it is not a function. By restriction the range of $\sqrt{x}$, every positive real number can be associated to one element in $\mathbb{R}$, which turns $\sqrt{x}$ into a function.