函数f(x)的定义域为R,且满足:f(x)是偶函数,f(x-1)是奇函数,若f(0.5)=9,则f(7.5)等于

问题描述:

函数f(x)的定义域为R,且满足:f(x)是偶函数,f(x-1)是奇函数,若f(0.5)=9,则f(7.5)等于

f(x)是偶函数得f(-x)=f(x)
f(x-1)是奇函数得f(-x-1)=-f(x-1),以x-1代换成x得f(-x-2)=-f(x)
于是f(x+2)=f(-x-2)=-f(x),f(x+4)=-f(x+2)=f(x),所以f(x)是周期为4的周期函数.

f(7.5)=f(-0.5)= f(0.5) =9