Let the sets A and B be two groups of non-empty numbers, then the mapping F:A→B from the set A to the set B is called the function f, and is denoted as y = f (x).

Where x is an independent variable and set a is a domain. The range of function value y is called range.

This is the mapping definition of the function. Redefined as follows.

The two sets a and b and the corresponding relation f are known.

If any element X in the set A has a unique element Y in the set B, it is called the mapping F from A to B. Note: F: A → B. Where element Y is the image of element X and element X is the original image of element Y. 。

By comparison, the relationship between function and mapping is obtained. In concept classification, mapping is a concept and function is a concept. The essence of a function is mapping.

Functions are special mappings. The particularity lies in: first, the collection is special. Is the mapping of two non-empty sets. Therefore, there is no function whose domain is an empty set. Second, the elements are special. Is a set of mappings of two numbers. The elements of these two sets can only be numbers. Can be a real number, before learning complex numbers; It can also be a plural number. Including imaginary numbers, of course.

When these two sets A and B are real numbers, they are called real variable functions. Function for short. It is the most basic and important function. It is also the most important function of middle school and university teaching in China at present.

When these two sets A and B are both complex numbers, they are called complex variable functions.

You are quite right: there is no one-to-many restriction in the broad definition of a function.

You are not right: one-to-many is also a function. It's called a multivalued function.

Why define one-to-many is not a function?

This restriction is only made within the scope of secondary schools.

Middle school mathematics education is a kind of basic education and universal education.

Middle school function has three limitations: real number range; One yuan; Single value. Strictly speaking, it is called a univariate single-valued real function.

This is the most basic, simple, easy to learn and widely used function in the salon.