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