Question

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

Answer 1

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

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

$\{so,putx\to \frac{1}{1-x}\}$

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

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

now, $f\left[f\left\{f\right(x\left)\right\}\right]=f\left[\frac{x-1}{x}\right]$

$\{so,putx\to \frac{x-1}{x}\}$

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

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

so, $f\left[f\left\{f\right(x\left)\right\}\right]=x$