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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?

symplectomorphic
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Coding247
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    Because the symbol $\sqrt{\cdot}$ is defined to mean "take the nonnegative square root of $\cdot$." Your first sentence is wrong. The solution set of $x^2=9$ is $\left{\pm3\right}$, but $\sqrt{9}=3$, period. – symplectomorphic Mar 08 '17 at 00:45
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    "It is definitely true" is always a bad argument. Most mathematicians define $\sqrt{\cdot}$ differently. – Thomas Andrews Mar 08 '17 at 00:47
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    In particular, functions are, by definition, single-valued. – Thomas Andrews Mar 08 '17 at 00:48
  • Also, the statement $\color{red}{\sqrt{25}=-5,5}$ is patently absurd. First, it is syntactically absurd: you can't list numbers using commas after equal signs. (When teachers write things like $x=1, 2$ they are being very naughty.) Second, it is semantically absurd: if $\sqrt{25}=5$ and $\sqrt{25}=-5$, then $5=-5$ (because $=$ is transitive). But $5$ isn't equal to $-5$. – symplectomorphic Mar 08 '17 at 00:50
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    The square root is different from the solution set of a quadratic. Also, if the range did include negative numbers, there will be two values of $y$ for one value of $x$. Since $f(x)=\sqrt x$ is a function, this is a contradiction. – tc216 Mar 08 '17 at 00:54
  • @symplectomorphic Everybody knows that $x=a,b$ means "$x=a$ or $x=b$." That is pedantic without being helpful. – Thomas Andrews Mar 08 '17 at 00:54
  • @ThomasAndrews: I disagree. It's not pedantic for an audience of middle or high school students. Writing $x=a, b$ can breed precisely the sort of confusion witnessed here. Your "everybody" means "everybody with enough training not to commit these kinds of elementary errors." But students do commit these errors. They aren't thinking about the semantics of the notation they use. – symplectomorphic Mar 08 '17 at 00:55
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    @ThomasAndrews So, would you say that it is definitely true that "'it is definitely true' is always a bad argument"? – Chris Mar 08 '17 at 00:55

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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 $\sqrt 9$.

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.

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

Michael McGovern
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