Advertisements
Advertisements
Question
If \[f\left( x \right) = \frac{1}{1 - x}\] , show that f[f[f(x)]] = x.
Advertisements
Solution
Given
\[f\left( x \right) = \frac{1}{1 - x}\]
Thus,
\[f\left\{ f\left( x \right) \right\} = f\left\{ \frac{1}{1 - x} \right\}\]
\[= \frac{1}{1 - \frac{1}{1 - x}}\]
\[= \frac{1}{\frac{1 - x - 1}{1 - x}}\]
\[ = \frac{1 - x}{- x}\]
\[ = \frac{x - 1}{x}\]
\[ = \frac{1 - x}{- x}\]
\[ = \frac{x - 1}{x}\]
Again ,
\[f\left[ f\left\{ f\left( x \right) \right\} \right] = f\left[ \frac{x - 1}{x} \right]\]
\[= \frac{1}{1 - \left( \frac{x - 1}{x} \right)}\]
\[ = \frac{1}{\frac{x - x + 1}{x}}\]
\[ = \frac{x}{1}\]
\[ = x\]
\[ = \frac{1}{\frac{x - x + 1}{x}}\]
\[ = \frac{x}{1}\]
\[ = x\]
Therefore, f[f{f(x)}] = x.
Hence proved.
Hence proved.
shaalaa.com
Is there an error in this question or solution?
