Question

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

Answer 1

Given:
$f\left(x\right)=\frac{x+1}{x-1}$
Therefore,
$f\left[f\left\{\left(x\right)\right\}\right]=f\left(\frac{x+1}{x-1}\right)$
$=\frac{\left({\displaystyle \frac{x+1}{x-1}}\right)+1}{\left({\displaystyle \frac{x+1}{x-1}}\right)-1}$
$=\frac{{\displaystyle \frac{x+1+x-1}{x-1}}}{{\displaystyle \frac{x+1-x+1}{x-1}}}=\frac{{\displaystyle \frac{2x}{x-1}}}{{\displaystyle \frac{2}{x-1}}}=\frac{2x}{2}=x$
Thus,
f [ f {(x)}] = x
Hence proved.