作业帮设偶函数f(x)对任意x∈R,都有,都有f(x+3)=-1/f(x),且当x∈[-3,-2]时,f(x)=4x,f(107.5)=?问：为什么f(107.5)=f(5.5)?
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/05 07:01:06
![设偶函数f(x)对任意x∈R,都有,都有f(x+3)=-1/f(x),且当x∈[-3,-2]时,f(x)=4x,f(107.5)=?问：为什么f(107.5)=f(5.5)?](/uploads/image/z/6937093-37-3.jpg?t=%E8%AE%BE%E5%81%B6%E5%87%BD%E6%95%B0f%28x%29%E5%AF%B9%E4%BB%BB%E6%84%8Fx%E2%88%88R%2C%E9%83%BD%E6%9C%89%2C%E9%83%BD%E6%9C%89f%28x%2B3%29%3D-1%2Ff%28x%29%2C%E4%B8%94%E5%BD%93x%E2%88%88%5B-3%2C-2%5D%E6%97%B6%2Cf%28x%29%3D4x%2Cf%28107.5%29%3D%3F%E9%97%AE%EF%BC%9A%E4%B8%BA%E4%BB%80%E4%B9%88f%28107.5%29%3Df%285.5%29%3F)
设偶函数f(x)对任意x∈R,都有,都有f(x+3)=-1/f(x),且当x∈[-3,-2]时,f(x)=4x,f(107.5)=?问：为什么f(107.5)=f(5.5)?
设偶函数f(x)对任意x∈R,都有,都有f(x+3)=-1/f(x),且当x∈[-3,-2]时,f(x)=4x,f(107.5)=?
问：为什么f(107.5)=f(5.5)?
设偶函数f(x)对任意x∈R,都有,都有f(x+3)=-1/f(x),且当x∈[-3,-2]时,f(x)=4x,f(107.5)=?问：为什么f(107.5)=f(5.5)?
f(x+3)=-1/f(x)
f(x+6)=-1/f(x+3)
所以f(x+6)=f(x)
所以f(107.5)=f(101.5)=f(95.5)=f(89.5)=……=f(5.5)
因为f(x)是偶函数,所以f(107.5)=f(5.5)=f(-5.5)
因为当x∈[-3,-2]时,f(x)=4x,所以f(-2.5)=-1/f(-5.5)=-10
所以f(107.5)=f(-5.5)=0.1
f(x+3)=-1/f(x)
f(x+6)=f[(x+3)+3]=-1/f(x+3)=f(x)
∴f(x)是周期函数，周期为6
∴f(107.5)=f(17×6+5.5)=f(5.5）
f(x+3)=-1/f(x),
则f(x+6)=f((x+3)+3)= -1/ f(x+3)= f(x).
所以函数周期为6
f(107.5)=f(17×6+5.5)= f(5.5)
= f(-0.5)
= -1/f(-0.5+3)=-1/f(2.5)
=-1/f(-2.5)
=-1/(4*(-2.5))
=1/10
f(x+3)=-1/f(x）
f(x+6)=f[(x+3)+3]=-1/f(x+3)=f(x)
所以T=6
当x∈[-3,-2]时，f(x)=4x
所以f(107.5)=f(17×6+5.5)=f(5.5）
因为f(x+6)=f[(x+3)+3)=-1/f(x+3)=-1/[-1/f(x)]=f(x)，
所以6是f(x)的一个周期，107.5=17*6+5.5, f(107.5)=f(5.5),
又f(5.5)=f(2.5+3)=-1/f(2.5)=-1/f(-2.5)=-1/-10=1/10