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

We have,
$f\left(x\right)=\frac{1}{1-x}$
Now,
$f\left\{f\right(x\left)\right\}=f\left\{\frac{1}{1-x}\right\}$
$=\frac{1}{1-\frac{1}{1-x}}$
$=\frac{1}{\frac{1-x-1}{1-x}}$
$=\frac{1-x}{-x}$
$=\frac{x-1}{x}$
$\therefore f\left[f\right\{x\left\}\right]=f\left\{\frac{x-1}{x}\right\}$
$=\frac{1}{1-\left\{\frac{x-1}{x}\right\}}$
$=\frac{1}{\frac{x-x+1}{x}}$
$=\frac{x}{1}$
$=x$
$\therefore f\left[f\right(x\left)\right]=x$
Hence, proved.