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I've got as part of an assignment to determine whether a given function is total, and if so, to say whether it's injective, surjective, or bijective. I can tell the answer by looking at it, but I feel that I need to make some kind of concrete proof that the function is total.

$f:\mathbb{N}\to \mathbb{N},f(x)=x^2+4$

I can tell by looking at it that it's a one-to-one total function. But is there a satisfactory way I can prove that this is a total function? I'm extremely new to this whole idea of functions outside of algebraic functions (such as total and partial functions)...

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    @agent154: If I write down $f(x)=\mbox{what wj32 says f(x) ought to be}$, is $f$ a total function? It is to me, but it might not be to you. You have to make sure what you write down is well-defined to people who read it, but looking for some kind of a formal proof that $f$ is a total function is unnecessary unless you are working in a computer algebra or automated reasoning system.2012-11-07

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I went with Brian's suggestion from above comments:

You have the idea. How about: Let $n\in \mathbb{N}$. $\mathbb{N}$ is closed under multiplication and addition, so $f(n)=n^2+4\in \mathbb{N}$. Therefore f is total.

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To formally prove that it's a total function, you need to look up the definitions/basic theorems about $\mathbb{N}$ and multiplication (or squaring) and addition, and check that

  1. $x^2+4$ has a fixed value for each $x\in\mathbb{N}$.
  2. That value is actually in $\mathbb{N}$. (This is something to check if your definitions only ensure it's in $\mathbb{Z}$, or something like that.)

Since I don't have your definitions, and you may not even have definitions for some of this stuff, I can't help you further except to say "if you don't have definitions for something, just argue intuitively for that part".

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    Well, you're probably not expected to worry about formally proving it completely rigorously at this level. But you seemed worried about rigor, so I described what more rigor would be like. If you feel this question is answered you can either accept my answer or add an answer of your own and accept that (since you had an answer first), or I think you can vote to delete your question if you really want.2012-11-13