f(n) = 2n is not surjective

Here is my first question, I'm preparing for the Analysis 1 exam and I found something that bothers me (not the first that shakes me I have to admit). If this is not a suitable question for mathoverflow just let me now, but I think that this can be of some help to people like me who are starting learning math.
I'm studying functions, and here is my problem:

$X = Y = {\mathbb N}$, and $f(n) = 2n$. Then I have $f({\mathbb N}) = 2{\mathbb N}$. If $m$ is even $f^{-1}(m) = {m/2}$, If m is odd $f^{-1}(m) = \emptyset$

The book I'm studying on says f is not surjective, how this can be?
I can say that the graphic of that function is as follows:

My only assumption is that:

$f(x) = y$ is not the same as saying $f(n) = n$

So I cannot use the graphic to understand the problem, but why is that?
By the way I think I'm still too focused on graphical representation of things, is this bad in math?

P.S.: Sorry for bad Latex formatting, this was my real first usage.

-
 Your f is not surjective because 1 is natural number, but being odd, it cannot be the image of any other number multiplied by 2 (as all of them are even). This being said, I do think this question is not fine for MathOverflow. As stated in the FAQ: "The primary criterion for determining whether a question is appropriate for Math Overflow is "is this of interest to mathematicians?" By a "mathematician," we mean a person whose primary occupation is doing mathematics." This question is more adecuate for general Math Forums. -1 – Jose Brox Nov 19 2009 at 17:44

Well, the question about whether graphics representations of things is helpful is kind of a mixed bag. It's not absolutely bad to draw a picture to help understand the problem - actually for some areas it could be an excellent idea depending on how you learn. However the problem in this case is that the picture is misleading. Your picture shows an image which is really defined on all the real numbers, but if I understand the question correctly - the function is on the counting numbers (natural numbers) (i.e. 1, 2, 3, 4, etc).

It seems to me that's where your problem stems from as well - you are looking it at like it's on all the real numbers, not just the counting ones.

If we recall the definition of surjective:

In mathematics, a function is said to be surjective or onto if its range is equal to its codomain. A function f: X → Y is surjective if and only if for every y in the codomain Y there is at least one x in the domain X such that f(x) = y. A surjective function is called a surjection.

(stolen from wikipedia so I don't have to type it myself).

Then X here is N and Y is N and there is no number n in N such that 2 * n = 1, thus so no value satisfies f(n) = 1 and the function is not surjective.

In regards to your last question - my opinion (I'm not a site admin or anything) is that this would qualify as a pretty good question for the site.

-
 Totally missed that I was working in the natural numbers. I really have to be more focused on that, I forget the domain more often than I like to admit; thank you streklin. – Montecristo Oct 4 2009 at 17:53 Yeah, this question is fine for mathoverflow. Also -- congratulations on posting the first question with embedded LaTeX! – Scott Morrison♦ Oct 4 2009 at 19:56

Thanks to the magic of tikz, pdfcrop, pdftoppm, and pnmtopng, here's the graph that you should have drawn.

-
That is nice, thank you. – Montecristo Oct 6 2009 at 7:42
May I ask, what you used to draw that graph? – Jose Capco Nov 19 2009 at 9:17
tikz! (The other programs mentioned were to convert the pdf to something that could be included here). It's a graphics package for LaTeX and is absolutely fantastic. It's in texlive so you should already have it. A really good site for finding out about it is texample.net. If you want the specific code for that graph, copy the link location into your browser and change the png extension to tex. – Andrew Stacey Nov 19 2009 at 10:19
This is a great example of tikz! Thanks for sharing. – Konrad Swanepoel Nov 19 2009 at 12:12