Domain: The set of values of independent variables is called the domain of a function.

Function value: the corresponding value of the independent variable is called the function value.

Range: The set of function values is called range.

From the concept of mapping, a function is actually a mapping from function set A to set B, where both A and B are non-empty number sets, and there is a unique function value corresponding to any value of the independent variable in the definition field A; The value of the independent variable is the original image, and its corresponding function value is the image; The set A of the original image is the definition domain, and the set C of the image is the range of the function. Obviously,

Function: If both A and B are non-empty number sets, then the mapping from A to B is called a function from A to B, and it is recorded as:.

note:

The three elements of the 1 function are the domain, the range and the corresponding rules.

2 Definition domain A= original image set, and value domain C = image set.

All elements in 3 C have original images, but they are not necessarily unique. Note the difference between them and one-to-one mapping.

4 A and b are two groups of non-empty numbers, so we should pay attention to their differences from a and b in the definition of mapping.

5 A function must be a mapping, and a mapping is not necessarily a function. As long as a and b are non-empty number sets, mapping is a function.

6 the independent variable takes any value in a, and the corresponding function value is.

When learning multiple functions at the same time, the symbol of the function can also be expressed by ….

Example: (1) linear function is the mapping from set A(A=R) to set B(B=R), so that the elements in set b correspond to those in set a, which is expressed as:, A= domain, and range c = b.

(2) The inverse proportional function is the mapping from set to set B(B=R), so that the elements in set B correspond to the elements in set A, which is denoted as:, A= definition domain, and the value domain C is the proper subset of B.

(3) The quadratic function is the mapping from set A(A=R) to set B(B=R), so that the elements in set B correspond to those in set A, and it is recorded as: A= domain, then, the range at that time (c is the proper subset of b).

(2) Representation method of functions:

There are three commonly used methods to express functions: analytical method, list method and image method.

(1) analytical method: an equation is used to express the functional relationship between two variables, which is called the analytical expression of the function. For example:

Advantages: the function relationship is clear, and it is easy to find the corresponding function value from the value of the independent variable, which is convenient for studying the properties of the function with analytical expressions.

(2) List method: use a list to represent the functional relationship between two variables. Such as: square table, square root table, trigonometric function table, interest table, etc.

Advantages: When the independent variable takes some values, you can know the corresponding value of the function without calculation.

(3) Image method: use the image of function to express the functional relationship between two variables. Such as: stock trend chart, automatic recorder applied by meteorological station, etc.

Advantages: Expressed the change of function intuitively.

(3) Interval:

Closed interval: A set satisfying real numbers is called a closed interval. []

Open interval: the set satisfying real numbers is called closed interval. ()

Semi-open and semi-closed interval: a set of real numbers that satisfies or is called a semi-open and semi-closed interval. or

Real number set r:

Pay attention to some appearances in the interval, such as.

(4) the domain of the function:

Domain: A collection of independent variables that make a function expression meaningful.

Note: A domain must be represented by a set.

Example: Find the domain of the following function:

①; ②; ③。

Summary:

(1) If it is an algebraic expression, then the domain of the function is the real number set r;

(2) If it is a fraction, then the domain of the function is the set of real numbers whose denominator is not equal to 0;

(3) If it is an even root, then the domain of the function is the set of real numbers whose modulus is greater than or equal to 0;

(4) If it is composed of several mathematical formulas, then the domain of the function is a group of real numbers that are meaningful by the mathematical formulas of each part.

(5) Method for judging whether two functions are the same function:

For example, which of the following functions is the same as the function?

①; ②; ③; ④。

Summary: Just look at whether the domain and the corresponding rules are the same.

(6) The function of image:

Example 1, the amount of money (yuan) for a certain kind of teacup per 5 yuan:,, Draw the image of this function.

Example 2. For mailing letters in China, assuming that each letter does not exceed 20g, pay 80 cents; if it exceeds 20g and does not exceed 40g, pay 160 cents, and so on, the postage payable for each letter is (unit: minutes):

Draw an image of this function.

(Diagram of Example 2) (Diagram of Example 3)

Note: The image of a function is usually one or several smooth curves, but sometimes it can be composed of some isolated points or several line segments.

Example 3. Draw an image of a function

Note: In its domain, a function has different corresponding rules for different ranges of independent variables. Such a function is called a piecewise function. Note that the piecewise function is one function, not several functions.