已知奇函数f(x)满足f(x+2)=f(x)对任意实数x都成立,且当x∈(0,1)时f(x)=2^x,则f[log0.5(23)]=同上

Question

已知奇函数f(x)满足f(x+2)=f(x)对任意实数x都成立,且当x∈(0,1)时f(x)=2^x,则f[log0.5(23)]=
同上

Answer 1

log0.5(23)=lg23/lg0.5
=lg23/(-lg2)
=-log2(23)
奇函数
f[log0.5(23)]=-f[log2(23)]
f(x+2)=f(x)
2是函数的周期
所以f(x)=f(x-2)
f[log2(23)]
=f[log2(23)-2]
=f[log2(23÷2²)]
=f[log2(5.75)]
=f[log2(5.75)-2]
=f[log2(5.75÷2²)]
=f[log2(1.4375)]
1<1.4375<2
所以0所以log2(1.4375)∈(0,1)
所以f[log2(1.4375)]=2^log2(1.4375)=1.4375
即f[log2(23)]=1.4375
所以f[log0.5(23)]=-1.4375