Question

If\[f\left( x \right) = 1 - \frac{1}{x}\] , then write the value of \[f\left( f\left( \frac{1}{x} \right) \right)\]

 

 

Answer 1

Given:  \[f\left( x \right) = 1 - \frac{1}{x}\]
Now,  \[f\left( \frac{1}{x} \right) = 1 - \frac{1}{\frac{1}{x}} = 1 - x\]

\[\Rightarrow f\left( f\left( \frac{1}{x} \right) \right) = f\left( 1 - x \right)\] Again,
 If \[f\left( x \right) = 1 - \frac{1}{x}\]

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

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