Scalars -> Vectors -> Matrices -> Tensors (really tensors?)
Assume f: X -> Y. We can now map x_1 to y_1 f(x_1)=y_1. And then change this same function by mapping x_1 to y_2: f(x_1)=y_2. Thus we can easily modify functions. Moreover, we do it constantly when we modify object fields in OOP. It is probably easier to comprehend if a function is represented as a table which we modify.
In contrast, we cannot modify data values (mathematical tuples). Say, x=42+1 means that a new value 43 is created rather than the existing value 42 is modified.
> I would expect the domain and codomain to be immutable;
No. Domains, codomains and any set can well be modified by adding or removing tuples. What is immutable are values (in the sets).