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Question
If \[f\left( x \right) = \frac{1}{1 - x}\] , show that f[f[f(x)]] = x.
Solution
Given
\[f\left( x \right) = \frac{1}{1 - x}\]
Thus,
\[f\left\{ f\left( x \right) \right\} = f\left\{ \frac{1}{1 - x} \right\}\]
\[= \frac{1}{1 - \frac{1}{1 - x}}\]
\[ = \frac{1 - x}{- x}\]
\[ = \frac{x - 1}{x}\]
\[ = \frac{1}{\frac{x - x + 1}{x}}\]
\[ = \frac{x}{1}\]
\[ = x\]
Hence proved.
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