There are six common forms of periodicity of functions: f (x+1) =-f (x) is one of them, and the method to prove the periodicity of such functions is always method of substitution (note: not method of substitution).

The process is as follows: if the condition f(x+ 1)=-f(x) (1) replaces x in the formula with x+ 1, we get:

f(x+ 1+ 1)=-f(x+ 1)(2)

Then bring f (x+1) =-f (x) in (1) to the right end of (2), and you can get:

f(x+ 1+ 1)=-f(x+ 1)=-(-f(x))= f(x)

Namely: f(x+2)=f(x), and the period T=2.

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