1.

If f (x) = 1/(1 – x), show that f [f {f (x)}] = x.

Answer»

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.



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