Then f (x+2a) = f [(x+a)+a] =-f (x+a) =-[-f (x)] = f (x)

So f(x) is a periodic function with a period of 2a.

2、f(x+a)= 1/f(x)

Then f (x+2) = f (x+)+one =1/f (x+a) =1/(1/f (x)) = f (x).

So f(x) is a periodic function with a period of 2a.

1、f(x+a)=- 1/f(x)

Then f (x+2) = f (x+)+one =1/f (x+a) =1/(1/f (x)) = f (x).

So f(x) is a periodic function with a period of 2a.

We got these three conclusions.

Extended data:

Important inference:

1. If the function f(x)(zhangx∈d) on shu field has two symmetry axes x = a and x = b, then the function f(x) is a periodic function, and the period t = 2 || b-a is the smallest (not necessarily a positive period).

2. If the function f(x)(x∈D) is located in the center of two symmetric domains A(A, 0), then the function B(B, 0) is a periodic function, and the period t = 2 || b-a is the minimum (not necessarily a positive period).

3. If the function f(x)(x∈D) and the domain of symmetry axis x = a and b are symmetrical about the center (b, 0) (representing b), then the function f(x) is a periodic function, and the period t = 4 | | B- is the lowest (not necessarily a virtuous cycle).

References:

Baidu Encyclopedia with Periodic Function