f(x)=(x-1/x+1)^2

1.y=f(x)=(x-1/x+1)^2=(x+1-2/x+1)^2=[(x+1/x+1)+(-2/x+1)]^2
=[1-(2/x+1)]^2

x>1 ====> x+1>2 ====> 0<(2/x+1)<1 ====> -1<-(2/x+1)<0
====> 0<1-(2/x+1)<1 ====> y^0.5=1-(2/x+1) ====> y^0.5-1=-2/x+1
====> x+1=2/(1-y^0.5) ====> x=[2/(1-y^0.5)]-1

交换x和y:f^-1(x)=[2/(1-x^0.5)]-1 (0<x<1)

2.设0<a<b<1
f^-1(b)-f^-1(a)=[(2/1-b^0.5)-1]-[(2/1-a^0.5)-1]
=2(1-a^0.5)-2(1-b^0.5)/(1-b^0.5)(1-a^0.5)
=2(b^0.5-a^0.5)/(1-b^0.5)(1-a^0.5)

0<a<b<1 ====> (1-a^0.5)>0,(1-b^0.5)>0,(b^0.5>a^0.5)>0
====> f^-1(x)>0 ====> f^-1单调增.

1. f-1(x)=[2/(1-x^0.5)]-1 0<x<1
2.单调递增