Question

m是非零常数,x∈R,且f(x+m)=(1+f(x))/(1-f(x)),问f(x)是否是周期函数?若是求出T,若不是,

Answer 1

周期为4m

f(x+2m)=f[(x+m)+m]=[1+f(x+m)]/[1-f(x+m)] ⑴

f（x+m）=[1+f(x)]/[1-f(x)] ⑵

由⑴⑵推出f(x+2m)=-(1/f(x)) ⑶

f(x+4m)=f[(x+2m)+2m]=-(1/f(x+2m))=f(x)

所以f(x)为周期函数,周期为4m