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+2a) = f [(x+a)+a] =1/f (x+a) =1[1/f (x)] = f (x).
So f(x) is a periodic function with a period of 2a.
3、f(x+a)=- 1/f(x)
Then f (x+2a) = f [(x+a)+a] =-1/f (x+a) =1[-1/f (x)] = f (x).
So f(x) is a periodic function with a period of 2a.
So we got these three conclusions.
Extended data
Important inference:
1. If the function f(x)(x∈D) has two symmetry axes in the domain, x=a and x=b, then the function f(x) is a periodic function, and the period T=2|b-a| (not necessarily the minimum positive period).
2. If the function f(x)(x∈D) has two symmetrical centers A(a, 0) and B(b, 0) in the definition domain, then the function f(x) is a periodic function, and the period T=2|b-a| (not necessarily the minimum positive period).
3. If the function f(x)(x∈D) has an axis of symmetry x=a and a center of symmetry B(b, 0)(a≠b), then the function f(x) is a periodic function with a period T=4|b-a| (not necessarily the minimum positive period).
References:
Baidu Encyclopedia with Periodic Function