Question

已知函数f(x)满足：f1=1/4.4f(x)·f(y)=f(x+y)+f(x-y)(x,y∈R),则f(2010)=?

Answer 1

令y=1,4f(x)f(y)=f(x+y)+f(x-y)得f(x)=f(x+1)+f(x-1) .(1) 由于x得任意性,用x+1代x ,所以 f(x+1)=f(x+2)+f(x) (2) (1)+(2) 有f(x+2)=-f(x-1),即f(x)=-f(x-3) f(2010)=f(0) 4f(x)f(y)=f(x+y)+f(x-y),令x=1,y=0,可得f(0)=0.5,所以f(2010)=0.5