For example, the following is a series of 2a- periodic functions.
F(x+a)=-f(x), so if f(x+a+a)=-f(x+a)=f(x), it will be decomposed into f(x)=f(x+2a). The key is to substitute with the overall idea.
Definition of function periodicity: If there is a constant t, f(x)=f(x+T) which is constant for any x in the defined domain, then f(x) is called a periodic function, and t is called a period of this function.
Extended data:
The key word of the function is "periodic repetition". When the independent variable increases to any real number (the independent variable is meaningful), the function value appears regularly and repeatedly.
If the function f(x)=f(x+T) (or f(x+a)=f(x-b) where a+b=T), then t is said to be a period of the function. An integer multiple of t is also the period of a function.
Show the definition of function periodicity: for the function y=f(x), if there is a non-zero constant t, which makes f(x+T)=f(x) hold when x takes any value in the defined domain, the function y=f(x) is called a periodic function, and the non-zero constant t is called the period of the function.
When the independent variable increases a certain value, the function value appears regularly.
2. Definition: For the function y=f(x), if there is a non-zero constant t, when x takes every value in the definition domain, f(x+T)=f(x).
The concretization of concepts:
When f(x)=sinx or cosx in the definition, the value of t is considered.
T=2kπ(k∈Z and k≠0)
So sine function and cosine function are both periodic functions, and the period is T=2kπ(k∈Z, k≠0).
Show pictures of sine and cosine functions.
The image shape of the periodic function changes periodically with the change of X. )
Emphasize the definition of "when x takes every value in the domain"
Let (x+T)2=x2, then x2+2xT+T2=x2.
So 2xT+T2=0, which means T(2x+T)=0.
So T=0 or T=-2x.
Emphasize "non-zero" and "constant" in the definition.
For example: trigonometric function sin(x+T)=sinx.
Cos (x+t) = t in cosx takes 2π.
3, the concept of minimum positive period:
For a function f(x), if all its periods have a minimum positive number, then this minimum positive number is called the minimum positive period of f(x).
For sine function y=sinx, as long as the independent variable x is at least increased to x+2π, the function value can be obtained repeatedly. So the minimum positive period of sine function and cosine function is 2π. (Note: Unless otherwise specified, period refers to the minimum positive period. )
On the function image, the minimum positive period is the shortest distance required for the function image to appear repeatedly.
References:
Baidu encyclopedia-function periodicity