Question

if f(x) =1/1-x show f(f(f(x)=x

Answer 1

Answer:

Given:

$f(x)=\frac{1}{1-x}$

To Show: f( f( f(x) ) ) = x

Consider,

LHS

= f( f( f(x) ) )

$=f(f(\frac{1}{1-x}))$

$=f(\frac{1}{1-\frac{1}{1-x}})$

$=f(\frac{1-x}{1-x-1})$

$=f(\frac{1-x}{-x})$

$=f(\frac{x-1}{x})$

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

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

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

$=x$

$=RHS$

Hence Proved.

Answer 2

Answer and explanation:

Given : $f(x)=\frac{1}{1-x}$

To show : $f(f(f(x)))=x$

Solution :

Taking LHS,

$f( f( f(x) ) )$

Substitute, $f(x)=\frac{1}{1-x}$

$=f(f(\frac{1}{1-x}))$

$=f(\frac{1}{1-\frac{1}{1-x}})$

$=f(\frac{1-x}{1-x-1})$

$=f(\frac{1-x}{-x})$

$=f(\frac{x-1}{x})$

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

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

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

$=x$

= RHS

Hence proved.