Every function has a domain, a codomain and an image, the last being a subset of the codomain.
My question, for which I ask for your opinions, is "Why is the domain not split into two parts in a similar fashion to the codomain?".
I believe that it's because there's no logical or analytical gain in doing so, but I will like a more concrete explanation. Or maybe to make deductions one needs to have one of the two sets (inputs and outputs) always being explicitly specified while the other one can vary.
Thank you for your time
Short answer: It's the definition of a function.
Long answer: Because a function maps everything in its domain to somewhere in the codomain - in order to define a function, you need to say where every point in its domain gets mapped to in its codomain, otherwise it's not a well-defined function. If a function only mapped some of the elements in its domain somewhere, then it would no longer be a function by definition. One usually does this by instead restricting the domain of a function - it's important to note though that this gives a different function to the original, as the new function will have different properties.
For instance, the function $f \colon \mathbb{R} \to \mathbb{R}$ given by $f(x)=x^2$ is not injective, however the function $g \colon [0,\infty) \to \mathbb{R}$ given by $g(x)=x^2$ is injective. In this particular case, $g$ is a restriction of $f$ to the subset of $f$'s domain, $[0,\infty) \subset \mathbb{R} = \operatorname{dom}f$. Sometimes we write $g = f|_{[0,\infty)}$.
As an aside, you may be interested in the notion of relation which generalises the notion of a function. Very briefly, if we identify a function $f \colon X \to Y$ with its graph, so $$f = \{(x,y) \in X \times Y \mid f(x)=y\} \subset X \times Y,$$ then a relation is just any subset of $X \times Y$.
A function $f \colon X \to Y$ can then be seen as a relation on $X \times Y$ such that for every $x \in X$, there exists a unique point $y \in Y$ such that $(x,y) \in f$ - this is often taken to be the definition of a function in set theory.