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