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We all know that for every real and positive number $N$ we have

$$\sqrt{N} = \pm a$$

for example $\sqrt{25} = \pm 5$.

Now: why the plot of the function $\sqrt{x}$ has only a positive part in the Cartesian plane? I have two results, plus and minus a certain number, so why the plot takes into account only the positive results?

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    $\begingroup$ No, $\sqrt{25}$ is not $\pm 5$; it's $5$, by definition. $\endgroup$ – David Mitra Nov 26 '15 at 17:58
  • $\begingroup$ You could plot $f(x)=-\sqrt x$ to get the negative branch. $\endgroup$ – JB King Nov 26 '15 at 18:21
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By definition $\sqrt{x}$ indicates the positive (or principal) square root of $x$. If we want indicate all the two root we have to write explicitly $\pm\sqrt{x}$.

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    $\begingroup$ Oh thank you!! That was a doubt I had since lots of time. Thank you! $\endgroup$ – Von Neumann Nov 26 '15 at 18:21
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Actually, the function $f(x)=\sqrt{x}$ usually refers to the principal square root which is defined as the nonnegative solution to $r^2=x$ for $x\ge0$.

While it's true that in general there are two solutions to the equation $r^2=x$, we really like single valued functions.

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  • $\begingroup$ Of course, a multi-valued function would be an oxymoron (i.e. not really a function). $\endgroup$ – John Joy Nov 27 '15 at 0:22
  • $\begingroup$ @JohnJoy Not true, this would still be a function, but on a different target set. $\endgroup$ – Did Nov 29 '15 at 9:03
  • $\begingroup$ @Did Of course, you are correct, if the elements of range of the function are themselves, sets (e.g. $\{-1,1\}\in S$). $\endgroup$ – John Joy Nov 29 '15 at 14:07
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The √ symbol is used to denote the principal square root of a number, i.e. the positive one.

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