I want to learn about functions, and I really can't understand the definition of a function, why it's called that, and why it's used so...

The traditional definition of a function:

Located in a certain process of change there are two variables x, y, if for every definite value of x within a certain range, y has a unique definite value corresponding to it, then y is said to be a function of x, x is called the independent variable.

We call the set of values taken by the independent variable x the domain of definition of the function, the values of y corresponding to the independent variable x are called the function values, and the set of function values is called the domain of the function.

Proximate definition of function:

Let A, B are non-empty sets of numbers, f: x→y is a correspondence law from A to B, then the mapping f: A→B from A to B is called a function, denoted as y=f(x), where x∈A, y∈B, the original set of images A is called the domain of definition of the function f(x), and the set of images C is called the domain of the value of the function f(x), it is clear that there are CB.

The symbol y=f(x) is the mathematical representation of "y is a function of x", which should be understood as follows:

x is the independent variable, which is the object on which the law is imposed; f is the corresponding law, which can be one or more analytical formulas, graphs, tables, or textual descriptions; y is the independent variable, which is a function of x when x is a permitted function, which is a function of x when x is a permitted function, which is a function of x when x is a permitted function. y is a function of the independent variable; when x is a specific value allowed, the corresponding y value is the function value corresponding to the value of the independent variable; when f is expressed analytically, the analytic formula is a functional analytic formula. y=f(x) is only a function symbol, not that "y is equal to the product of f and x", and f(x) is not necessarily an analytic formula, in the study of functions, in addition to using the symbol f(x), it is not necessarily an analytic formula. f(x) is not necessarily an analytic formula, in the study of functions, in addition to the symbol f(x), but also commonly used g(x), F(x), G(x) and other symbols to indicate.

Understanding of the concept of function

The essence of the two definitions of function is the same, only the starting point of the concept of the narrative is different, the traditional definition is from the point of view of the change of the movement, while the modern definition is from the point of view of the set, mapping. In this way, it is not difficult to know that the function is essentially a special mapping from the set of non-empty numbers A to the set of non-empty numbers B.

From the recent definition of function, the concept of function contains three elements: the definition domain A, value domain C and the correspondence law f. The core of which is the correspondence law f, which is the essential feature of the functional relationship. y = f (x) means that y is equal to the corresponding value of x under the law f, and f is the method of the realization of "correspondence" and the way, is the link between x and y. f is the method and way in which the "correspondence" is realized, and it is the link between x and y, so it is the core of the function. It does not matter what letters are used to represent the independent variable, the dependent variable, and the correspondence rule.