Generally, Y is equal to C. For a function, if there is a non-zero constant that makes FX+T equal to FX when taking every value in the definition field, then the function Y is equal to F, which is called a periodic function and a non-zero constant.

It is called the period of this function. For a periodic function, if all periods have a minimum positive number, it is called the minimum positive period.

Properties of periodic functions:

1, if T(≠0) is the period of f(x), then -T is also the period of f(x).

2. If T(≠0) is the period of f(x), then nT(n is an arbitrary non-zero integer) is also the period of f(x).

3. If T 1 and T2 are both periods of f(x), then T 1 T2 is also a period of f(x).

4. If f(x) has a minimum positive period t, then any positive period t of f(x) must be a positive integer multiple of t. ..

5. If T 1 and T2 are two periods of f(x) and T 1/T2 is an irrational number, then f(x) has no minimum positive period.

6. The domain m of the periodic function f(x) must be an unbounded set with at least one side.

Representation of function:

Analytical formula method:

The method of expressing the functional relationship between two variables with an equation containing mathematical relationship is called analytical method. The advantage of this method is that it can express the quantitative relationship between function and independent variable concisely, accurately and clearly. The disadvantage is that it is often complicated to calculate the corresponding value, and some functional relationships in practical problems may not be expressed by expressions.

List method:

The method of expressing the functional relationship between two variables by list method is called list method. The advantage of this method is that the corresponding function values can be read directly by knowing the values of the independent variables in the table; The disadvantage is that only some corresponding values can be listed and it is difficult to reflect the whole picture of the function.

method of images

Take the values of the independent variable X and the corresponding dependent variable Y of a function as the abscissa and ordinate of a point respectively, and draw its corresponding point in the rectangular coordinate system. A graph composed of all these points is called an image of a function. This method of expressing functional relations is called image method.

Language narrative method:

Describe the relationship between functions in language.