Prove that a function is a total function (as opposed to a partial function)

Question

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)...

Answer 1

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.

Answer 2

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".