# Why $f(x) = \sqrt{x}$ is a function?

Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a rule of corre- spondence that assigns to each element in set A exactly one element in B.", so $f(x) = \sqrt{x}$ is not a function?

• @hippietrail, please read the description of the roots tag. It is not relevant to questions involving radicals. Feb 9, 2014 at 6:14

I will assume that in this portion of your textbook it is assumed that $x \in \mathbb{R}$, and with that condition $f(x)=\sqrt{x}$ is certainly a function. Specifically $f:[0,\infty) \rightarrow [0,\infty)$. It meets the formal definition of a function (not one to many).

Your confusion is due to an inappropriate extrapolation of reasoning. Specifically this:

You know that $x^2=25 \Rightarrow x= \pm \sqrt{25}$ by the square root property. However this function in no way involves taking the square root of both sides of an equality. It is just a function $f(x)=\sqrt{x}$, and the domain is $x \geq 0$ by virtue of the fact that you are living in the real number system in this portion of your textbook.

Now $f(x)= \pm \sqrt{x}$ is certainly not a function. For example, if the question were "Let $y^2=x$. Is $y$ a function of $x$?" You would say no in this case as $y= \pm \sqrt{x}$, and you would choose $x=25$ to counter the definition of a function.

Note, your statement "the square root of $25$ has two different outputs" is false. There is only one output. However, if $y^2=25$, then $y$ has two different solutions.

For functions defined by equations, we agree on the following convention regarding the domain: Unless otherwise indicated, the domain is assumed to be the set of all real numbers that lead to unique real-number outputs. The symbol $\displaystyle \sqrt{ }$ is defined in algebra to mean the positive square root only. Thus $\displaystyle \sqrt{25} = 5$, and if you are thinking of the other root, you need to write $\displaystyle -\sqrt{25} = -5$. Consequently, the function $\displaystyle f(x) = \sqrt{x}$ do represent a function; for each input there is exactly one output.

• That's problematic. What if there is no order, no canonical choice, for example in $\mathbb{C}$? For real numbers $x$, $\sqrt{x}$ customarily means the non-negative square root, but there is no law, one can follow other conventions. Sep 13, 2013 at 15:53
• Wait... why did you answer your own question? Sep 13, 2013 at 16:04
• @DanielFischer For functions defined by equations, we agree on the following convention regarding the domain: Unless otherwise indicated, the domain is assumed to be the set of all real numbers that lead to unique real-number outputs. So there must be a convention regarding $\sqrt{}$ and it was chosen to be a positive square root only for the set of all real numbers that lead to unique real-number outputs when dealing with functions.
– user93957
Sep 13, 2013 at 16:04
• @kaine Q/A style, share my own knowledge.
– user93957
Sep 13, 2013 at 16:04
• I am puzzled by your claim that “Unless otherwise indicated, the domain is assumed to be the set of all real numbers that lead to unique real-number outputs.”. I am not familiar with that convention, and it does not seem to make very much sense or to be consistent with what I already know. I worry that you made it up. Can you provide a source?
– MJD
Sep 13, 2013 at 16:38