证f(x+y)+f(x-y)=2f(x)f(y)为偶函数能否取-y?
问题描述：
证f(x+y)+f(x-y)=2f(x)f(y)为偶函数能否取-y?
f(x+y)+f(x-y)=2f(x)f(y)取-y即把y换成-y
f(x+y)+f(x-y)=2f(x)f(-y)比较两式得f(y)=f(-y)所以为偶函数?
那为什么书上不这么证
答
可以,已知等式f(x+y)+f(x-y)=2f(x)f(y),取-y后,得又一等式
f(x-y)+f(x+y)=2f(x)f(-y),
则由两等式左侧相等得到等式右侧也相等,
即为2f(x)f(y)=2f(x)f(-y),即f(y)=f(-y),所以此函数为偶函数.
证明完毕