Given as
f(x) = (x + 1)/(x – 1)
Let us prove that the f [f (x)] = x.
f [f (x)] = f [(x+1)/(x-1)]
= [(x+1)/(x-1) + 1]/[(x+1)/(x-1) – 1]
= [[(x+1) + (x-1)]/(x-1)]/[[(x+1) – (x-1)]/(x-1)]
= [(x+1) + (x-1)]/[(x+1) – (x-1)]
= (x+1+x-1)/(x+1-x+1)
= 2x/2
= x
∴ f [f (x)] = x
Thus proved.
