f(x)的反函数f^-1=log2(1+x)/(1-x),求f(x)的解析式和解不等式1-f(x)>1/(4^x-1)

题目

f(x)的反函数f^-1=log2(1+x)/(1-x),求f(x)的解析式和解不等式1-f(x)>1/(4^x-1)

答案

由f^-1=log2(1+x)/(1-x)可得
2^x=(1+y)/(1-y)
化简得
f(x)=(2^x-1)/(2^x+1)
1-(2^x-1)/(2^x+1)>1/(4^x-1)
等价于2/(2^x+1)>1/(2^x+1)(2^x-1)
2^x+1>0,则(2^x-3/2)/(2^x-1)>0
即2^x>3/2或0小于2^x小于1
则解集为{x/x>log2(3/2)或x小于0}