Let us prove that f [f {f (x)}] = x.
Firstly, let us solve for f {f (x)}.
f {f (x)} = \( \dfrac{1}{(1 – x)}\)
= \( \dfrac{ 1 }{ 1 }– \dfrac{1}{(1 – x)}\)
= \(\dfrac{ (1 – x) }{-x}\)
= \( \dfrac{ (x-1) }{x}\)
f {f (x)} = \( \dfrac{ (x-1) }{x}\)
Now, we shall solve for f [f {f (x)}]
f [f {f (x)}] = f [\( \dfrac{ (x-1) }{x}\)]
= \(\dfrac{1 }{ \dfrac{(x – x + 1)}{x}}\)
=\(\dfrac{ 1 }{ (\dfrac{1}{x})}\)
f [f {f (x)}] = x
Hence proved.
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