I think modern mathematics calls the corresponding rule F a function, which makes the concept of function more rigorous. Because the traditional function definition generally explains the relationship between X and Y, but this relationship may be incomplete (such as the correlation between variables). The correspondence law points out the definite relationship between two variables, the essence of which is that the dependent variable changes with the independent variable and is definite.

As for the function of calling the corresponding law f a function and y a function of x, it is the same. Take uniform linear motion as an example. S is a function of t because S=vt. S=vt is the corresponding relationship between s and t, so these two expressions are the same. (This relationship S=S(t) can be expressed by the definition of function: S: T→ VT, the mapping from time to displacement. And the expression S(t)=vt written at ordinary times is the value of the function s at t)

This is a completely understandable thing. It can be said that it can only be understood but not expressed. You have a good grasp.