In high school, we have learned the concepts of mapping , the concept of functions , the three representation methods of functions , the domain of functions , the independence of function equality and independent variable representation, etc. The function concepts involved in advanced mathematics or calculus courses are basically the **same as those learned in high school**.

**A function **is a mathematical model that describes the interdependence between variables.

In a certain natural phenomenon or social phenomenon, there are often multiple constantly changing quantities, that is, variables . These variables do not change in isolation, but are interconnected and follow certain laws. A function is a law that describes this connection. In this section we first discuss the case of two variables (the case of more than two variables will be discussed in the next volume) .

**For example**, in free fall motion, let the time of falling of the object be and the distance of falling be . Assuming that the moment when the fall begins is , the dependence between variables and is determined by the mathematical model

Given, where is the acceleration due to gravity.

** Definition: **Let and be two variables , be a given non-empty set of numbers . If for each number , according to a certain rule , there is always a certain value corresponding to the variable , then is said to be **a function **of , denoted as

.

Among them, is called **the independent variable**, is called **the dependent variable**, and the number set is called **the domain **of the function , also recorded as , that is

.

For , according to the corresponding rule , there is always a certain value ( marked as ) corresponding to it, and is called **the function value **of the function at point . This dependence between the dependent variable and the independent variable is often called **a functional relationship**.

When the independent variable takes all the values of , the set of all corresponding function values is called **the value range **of the function , denoted as or , that is

.

**Note 1: **The domain of a function and the corresponding rules are called **the two elements of the function**. The necessary and sufficient condition for two functions to be equal is that their **definition domains and corresponding rules are the same**, and it does not matter what letters are used to represent the independent variables of the function. This is **the irrelevance of the independent variables of the function**.

Regarding the domain of a function, it should be specifically determined based on the actual meaning of the problem in actual problems. If the discussion is about pure mathematics, the set of all real numbers that make the expression of the function meaningful is often taken as the domain of the function. This domain is also called the **natural domain** of the function .

**Note 2**：The domain of commonly used basic elementary functions .

**For example**, function

The (natural ) domain of is the open interval .