Given as
f (x) = 1 / (1 – x)
Let us prove that the f [f {f (x)}] = x.
Firstly, let us solve for the f {f (x)}.
f {f (x)} = f {1/(1 – x)}
= 1/1 – (1/(1 – x))
= 1/[(1 – x – 1)/(1 – x)]
= 1/(-x/(1 – x))
= (1 – x)/-x
= (x – 1)/x
∴ f {f (x)} = (x – 1)/x
Then, we shall solve for the f [f {f (x)}]
f [f {f (x)}] = f [(x-1)/x]
= 1/[1 – (x-1)/x]
= 1/[(x – (x-1))/x]
= 1/[(x – x + 1)/x]
= 1/(1/x)
∴ f [f {f (x)}] = x
Thus proved.
