The traditional definition of (1) function: it is assumed that there are two variables x and y in a certain change process. If Y has a unique fixed value corresponding to each fixed value of X within a certain range, Y is said to be a function of X, and X is called an independent variable.

(2) Modern definition of function: Let A and B be sets of non-empty numbers, and f:x→y be corresponding rules from A to B, then the mapping f:A→B from A to B is called a function, and it is denoted as Y = f(x), where x∈A, y∈B and the original image set A are called a function F (.

The above two definitions are essentially the same, but the traditional definition is based on the viewpoint of movement change, while the modern definition is based on the viewpoint of set and mapping, with different emphasis. Function is essentially a special mapping from set A to set B, which is unique in that both sets A and B are non-empty sets. The set of independent variables is called the domain of function, and the set of function value c is called the domain of function.

It should be noted that the range C is not necessarily equal to the set B, only that C is a subset of B. 。

2. The three elements of a function

The definition field A, the value field C and the corresponding rule F from A to C are called the three elements of a function. Because the range can be uniquely determined by the definition field and the corresponding rules, it can also be said that the function has two elements: the definition field and the corresponding rules. Two functions are the same if and only if the domain and corresponding rules are the same respectively. This function represents a correspondence, in which each input value corresponds to a unique output value. The standard symbol of the output value corresponding to the input value x in the function f is f(x). The set containing all the input values of a function is called the domain of this function, and the set containing all the output values is called the range. If you define the concept of mapping first, you can simply define a function as a function, and the mapping defined between non-empty number sets is called a function, which is a special mapping.