Question

已知定义在R上的奇函数f(x)满足f(x+1)=-f(x),若f(1.5)=1,则f(2010.5)=?

Answer 1

因为f(x+1) = -f(x), 所以f(x+2) = f( (x+1)+1 ) = -f(x+1),

联立以上两式得f(x) = f(x+2) , 故f(x)为周期函数,T=2,

所以f(0.5) = f(0.5+2) = f(0.5+2) = f(0.5+2+2) = f(0.5+2+2+2) =.

= f(1.5+2*1005) = f(2010.5),

因为f(0.5+1) = -f(0.5) = 1, 所以f(0.5) = -1

且f(x)为奇函数, 即 -f(-x) = f(x),

所以 -f(-0.5) = f(0.5) = f(2010.5) = -1.

望采纳~