f(x+a)=-f(x).

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

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

f(x+a)= 1/f(x).

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

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

f(x+a)=- 1/f(x).

Then f (x+2a) = f =-1/f (x+a) =1[-1/f (x)] = f (x).

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

Source of function:

The word "function" used in China's mathematics books is a translated name, which was translated into "function" by Li, a mathematician in Qing Dynasty, when he translated Algebra (1859).

In ancient China, the word "Xin" and the word "Han" were universal and both had the meaning of "Han". The definition given by Li is: "Every formula contains days as a function of days." In ancient China, four different unknowns or variables were represented by four words: heaven, earth, people and things. The meaning of this definition is: "Whenever a formula contains a variable X, the formula is called a function of X."

So "function" means that the formula contains variables, and the exact definition of the equation means that it contains unknowns and so on. However, in the early mathematical monograph "Nine Chapters Arithmetic" in China, the word equation means simultaneous linear equations with many unknowns, that is, the so-called linear equations.