设偶函数f(x)对任意x∈R,都有f(x+3)=-1f(x),且当x∈[-3,-2]时,f(x)=4x,则f(107.5)=(  ) A.10 B.110 C.-10 D.-110

问题描述:

设偶函数f(x)对任意x∈R,都有f(x+3)=-

1
f(x)
,且当x∈[-3,-2]时,f(x)=4x,则f(107.5)=(  )
A. 10
B.
1
10

C. -10
D. -
1
10

因为f(x+3)=-

1
f(x)
,故有f(x+6)=-
1
f(x+3)
=-
1
1
f(x)
=f(x).函数f(x)是以6为周期的函数.
f(107.5)=f(6×17+5.5)=f(5.5)=-
1
f(2.5)
=-
1
f(−2.5)
=-
1
4×(−2.5)
=
1
10

故选B