Find the inverse of #f(x) = (x-1)/(x+1)# and then find #(f@f^-1)(x)#?

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Answer 1 · 2018-06-20T23:01:17

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

#(f@f^-1)(x)=x#

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

Let #y=f(x)#

#y=1-2/(x+1)#

#2/(x+1)=1-y#

#2=(1-y)(x+1)#

#2=x(1-y)+(1-y)#

#x(1-y)=2+y-1=y+1#

#x=(y+1)/(1-y)#

Thus #f^-1(x)=(x+1)/(1-x)#

By definition, we know that #(f@f^-1)(x)=x#