已知函数f(x)满足f(x+y)+f(x-y)=2f(x)×f(y).(x∈R,y∈R.),且f(0)≠0.(1)求f(0)=?已知函数f(x)满足f(x+y)+f(x-y)=2f(x)×f(y).(x∈R,y∈R.),且f(0)≠0.(1)求f(0)=?(2)证明f(x)是偶函数.

Question

已知函数f(x)满足f(x+y)+f(x-y)=2f(x)×f(y).(x∈R,y∈R.),且f(0)≠0.(1)求f(0)=?
已知函数f(x)满足f(x+y)+f(x-y)=2f(x)×f(y).(x∈R,y∈R.),且f(0)≠0.
(1)求f(0)=?
(2)证明f(x)是偶函数.

Answer 1

(1)因为f(x+y)+f(x-y)=2f(x)×f(y),所以f(0+0)+f(0-0)=2f(0)×f(0),又因为f(0)≠0
所以有f(0)=1;
(2)f(-x)+f(x)=f(-x+0)+f(x+0)=2f(x)f(0)=2f(x),所以有f(-x)＝f(x)从而f（x）为偶函数.