Question

If  \[f\left( x \right) = \frac{x + 1}{x - 1}\] , show that f[f[(x)]] = x.

 

 

Answer 1

Given:

\[f\left( x \right) = \frac{x + 1}{x - 1}\]

Therefore,

\[f\left[ f\left\{ \left( x \right) \right\} \right] = f\left( \frac{x + 1}{x - 1} \right)\]

\[= \frac{\left( \frac{x + 1}{x - 1} \right) + 1}{\left( \frac{x + 1}{x - 1} \right) - 1}\]

\[= \frac{\frac{x + 1 + x - 1}{x - 1}}{\frac{x + 1 - x + 1}{x - 1}} = \frac{\frac{2x}{x - 1}}{\frac{2}{x - 1}} = \frac{2x}{2} = x\]

Thus,
f [ f {(x)}] = x
Hence proved.