Show that \(f(f(f(x)))=x\)

Solution

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

First find \(f(f(x))\):

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

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

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

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

Now find \(f(f(f(x)))\):

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

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

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

$$ =x $$

Hence, $$ \boxed{f(f(f(x)))=x} $$