Function: In the field of mathematics, a function is a relationship that makes each element in one set correspond to the only element in another (possibly the same) set.

Understanding function can be understood from the following aspects:

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Two groups of elements

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Corresponding rules,

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Each element in the first group has only a unique corresponding number in the second group.

Terms: function, mapping, correspondence and transformation usually have the same meaning. But the function only represents the correspondence between numbers, and the mapping can also represent the correspondence between points and between graphs. It can be said that the mapping contains functions.

In short, a function is a "rule" that assigns a unique output value to each input. This "rule" can be expressed by a function expression, a mathematical relationship or a simple table listing input values and output values. The most important property of a function is certainty, that is, the same input always corresponds to the same output (note that the opposite is not necessarily true). From this point of view, a function can be regarded as a "machine" or a "black box", which converts a valid input value into a unique output value. Generally, the input value is called the parameter of the function, and the output value is called the value of the function.

The parameters and function values of the most common functions are numbers, and their corresponding relations are expressed by function expressions. The function values can be obtained by directly substituting the parameter values into the function expressions.

The concept of function is not limited to the calculation of numbers, or even to calculation. The mathematical concept of function is broader, which includes not only the mapping relationship between numbers. This function links the definition domain (input set) with the mapping domain (possible output set) so that each element of the definition domain uniquely corresponds to an element in the mapping domain. As described below, functions are abstractly defined as explicit mathematical relationships. Because of the generality of function definition, the concept of function is very basic for almost all branches of mathematics.