# How to prove this Surjectivity

How can I prove that this function is surjective and monotonic?

$$f:\left]-5,5\right[\to \left[0,1\right], \ x\mapsto e^{-x^2}$$

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What is $f(-\pi)$? –  Hagen von Eitzen Feb 3 '13 at 15:33
Does there exist a preimage of 0? –  Abhra Abir Kundu Feb 3 '13 at 15:34
sorry the function is $x\rightarrow e^{-x^2}$ –  Devid Feb 3 '13 at 15:36

First of all, it isn't monotonic (as it is an even, non-constant function). Second of all, at no point is $f(x)=0$, so it wouldn't be surjective as stated.

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sorry the function is $x\rightarrow e^{-x^2}$ –  Devid Feb 3 '13 at 15:37
That takes care of a few problems, but not all of them. See my edit. –  Cameron Buie Feb 3 '13 at 15:40
Ok but you proved that the function is not surjective because $f(x)\neq 0$. But how do you know that it is not monotonic, how do you prove that ? –  Devid Feb 3 '13 at 15:47
Note that $f(0)=1$, then take any $0<x<5$ and show that $f(-x)=f(x)<1$. Either that, or prove the more general result that no even, non-constant function on an open interval about $x=0$ can be monotonic. –  Cameron Buie Feb 3 '13 at 15:48
@Devid: Yes, that's certainly true. We generally only refer to a function as "monotonic" if it is increasing on its whole domain or decreasing on its whole domain. I see what you meant, now, though. –  Cameron Buie Feb 4 '13 at 2:34