已知函数f(x)=f(1/x)·lgx+1,则f(100)=

题目

已知函数f(x)=f(1/x)·lgx+1,则f(100)=

答案

f(100)=f(1/100)·lg100+1
=2f(1/100)+1
=2[f(100)·lg(1/100)+1]+1
=2[f(100)·(-2)+1]+1
令f(100)=t
则t=2[(-2)t+1]+1
解得t=3/5
所以f(100)=3/5