Correct way to formally prove whether a composite function exists or does not exist

Question

I am confident that my answer is correct, but I am not sure if the manner in which I have proved it is formal enough.

g(-1,-2) = (1/2 * 1, -2) = (1/2,-2)

x = 1/2 is within the parameter of the codomain of g in which { k/2 : k ∈ N}

y = -2 is within the parameter of the codomain of g in which -2 ∈ Z

f ◦ g((-1,-2)) = (-2, 1/2)

y = 1/2 is not within the parameter of the codomain of f as y is not an integer

Therefore, f ◦ g((-1,-2)) does not exist

How can I make my proof more formal?

Answer 1

The essence is to check if the codomain of $g$ is contained in the domain of $f$ and this should be stated in the proof. For instance:

In order to build the composite $f\circ g$ of two functions we have to assure that the codomain of $g$ is contained in the domain of $f$.

Here we have as codomain of $g$ the set $$\left\{\frac{k}{2}:k\in\mathbb{N}\right\}\times \mathbb{Z}$$ which is not contained in the domain $\mathbb{Z}\times\mathbb{Z}$ of the function $f$. Therefore the composite $f\circ g$ does not exist.

This can be seen when considering the point $(-1,-2)$. The function $g((x,y))=\left(\frac{1}{2}\left|x\right|,y\right)$ maps the point $(-1,-2)$ to $\left(\frac{1}{2},-2\right)$ which is not element of the domain $\mathbb{Z}\times\mathbb{Z}$ of the function $f$.