What is the difference between the traditional definition and the modern definition of function?

In the18th century, the traditional definition of function is:

[Algebraic expression with unknowns and constants]

.

From19th century till now, the modern definition of function is:

[Mapping relationship between two sets]

.

Therefore, the modern definition is broader than the traditional definition;

For example, the traditional function can only be an algebraic expression of f (x) = x 3+2x+ 1, and its input and output are all numbers, and the calculation process is only an algebraic algorithm;

But modern functions are different, such as:

Let A be a matrix, then "the transposition of f(A) = A" is a function, and its input and output are both matrices, not single numbers; And its calculation process does not include any algebraic algorithm;