Question

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

Answer 1

Given:
$f\left(x\right)=\frac{1}{1-x}$
Thus,
$f\left\{f\left(x\right)\right\}=f\left\{\frac{1}{1-x}\right\}$

$=\frac{1}{1-{\displaystyle \frac{1}{1-x}}}$

$$\begin{gathered} =\frac{1}{{\displaystyle \frac{1-x-1}{1-x}}} \\ =\frac{1-x}{-x} \\ =\frac{x-1}{x} \end{gathered}$$
Again,
$f\left[f\left\{f\left(x\right)\right\}\right]=f\left[\frac{x-1}{x}\right]$
$$\begin{gathered} =\frac{1}{1-\left({\displaystyle \frac{x-1}{x}}\right)} \\ =\frac{1}{{\displaystyle \frac{x-x+1}{x}}} \\ =\frac{x}{1} \\ =x \end{gathered}$$

Therefore, f[f{f(x)}] = x.
Hence proved.