设函数y=f(x)满足f(x+2)=-f(x),且f(2)=2求f(-2),f(4),f(100)的值

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设函数y=f(x)满足f(x+2)=-f(x),且f(2)=2求f(-2),f(4),f(100)的值
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设函数y=f(x)满足f(x+2)=-f(x),且f(2)=2求f(-2),f(4),f(100)的值
设函数y=f(x)满足f(x+2)=-f(x),且f(2)=2
求f(-2),f(4),f(100)的值

设函数y=f(x)满足f(x+2)=-f(x),且f(2)=2求f(-2),f(4),f(100)的值
解由f(x+2)=-f(x),
知f(x+4)
=f(x+2+2)
=-f(x+2)
=-[-f(x)]
=f(x)
即f(x+4)=f(x)
故f(x)的周期为4
则f(-2)=f(-2+4)=f(2)=2
又由f(x+2)=-f(x),
令x=0
则f(2)=-f(0)
则f(0)=-f(2)=-2
即f(4)=f(0)=-2
f(100)=f(25×4+0)=f(0)=-2

f(x+4)=-f(x+2)=f(x)
因此周期是4
f(-2)=f(2)=2
f(x+2)=-f(x)
x=0代入得
f(0)=-f(2)=-2=f(4)=f(100)