Why doesn't the range of $f(x)=\sqrt{x}$ include negative numbers?

Question

It is definitely true that $\sqrt{9}$ is both $3$ and $-3$, and that $\sqrt{25}=-5,5$.

When I graph the function $f(x)=\sqrt{x}$ its range is shown to be $y\geq0$.

Why doesn't its range include negative numbers?

Answer 1

No, it is not "definitely true that $\sqrt 9$ is both $3$ and $-3$".

As the symbol is usually defined, $\sqrt 9$ means the non-negative number whose square is $9$. In other words, it means $3$.

$-3$ is a number whose square is $9$, but because $-3$ fails to be a non-negative number, it is not the value of $\sqrt 9$.

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There are a few contexts where the $\sqrt{\cdots\vphantom x}$ notation is used not to denote a single number but the entire set of numbers whose square is "$\cdots$". Foremost this can be the case when complex numbers are involved. But this is still not the usual meaning of the symbol.

Answer 2

A fairly elementary reason is that $f(x)=\pm\sqrt{x}$ is not a function.