If f(x)
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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