Is a function onto if the output is a subset of the domain?

Question

1. $f : \mathbb{Z} \rightarrow \mathbb{Z}, f(x)= 2x+1$

2. $f : \mathbb{R} \rightarrow \mathbb{R}, f(x)= 2x+1$

I was told that 1) is 1-1, but not onto because the image only contains odd integers, whereas 2) is a Bijection.

Why is this so? Isn't a domain of odd integers a subset of all integers? i.e. if x = 3, the function would spit out 7 which is an odd integer.

Also:

3. $f : \mathbb{Z} \rightarrow \mathbb{R}, f(n)= \pm n$

This is not a function as a single input will yield two values, namely a positive and negative value.

However,

4. $f : \mathbb{Z} \rightarrow \mathbb{R}, f(n)= n$

Is this a function? Considering that any integer will yield another integer, and since integers are a subset of real numbers, is this a valid function?

Answer 1

To the first question, a surjective function explicitly requires that it covers the entire codomain, not just a subset. If the definition allowed for mappings to a subset of the codomain, then every function is surjective, and the term is meaningless. Yes, $f(3) = 7$, but for what value of $x$ does $f(x)=10$?

As for your second function, why do you think it wouldn't be a valid function? A function simply requires that every member of the domain has an image, and only one image. What has you concerned here?

Answer 2

Definition of a function has 3 ingredients, the domain set, the codomain set and the rule that associates to each element of the domain an element in the codomain. For a function written $g\colon X\to Y$, the domain is $X$, the codomain is $Y$.

A function is called onto, if by this rule, every element of the codomain is associated to some element of the domain.

So the function M: Humans .---> Humans, M(x) = mother of x is not onto because, Donald Trump (an element of the codomain) is not the mother for anyone (in the domain, or for that any set)

The case(i) the domain and codomain are integers. 10 is not obtainable as $2x+1$ for any x in the given domain (consisting of integers), so not onto. But in case (2) 10 is obtainable as 2x+1 for $x=4.5$ which is in the given domain of real numbers.