Question

If $f(x) = \frac{x+1}{x-1} , x \neq 1$ then (fofofof) is equal to

• A. x
• B. $\frac{1}{x}$
• C. $x^2$
• D. indeterminate

Answer 1

Answer: (A)

$\left(fof\right) x = f \left(\frac{x+1}{x-1}\right) = \frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}$
$ = \frac{x+1+x-1}{x+1-x+1}$
$ = \frac{2x}{2}=x$
Now,
$\left(fo fof\right)\left(x\right)=f\left[f\left\{f\left(x\right)\right\}\right] = f\left(x\right) = \frac{x+1}{x-1}$
Now, $ \left(fo fo fof\right)\left(x\right)=f\left[f\left\{f\left(x\right)\right\}\right] = f\left\{f\left(x\right)\right\} $
$= \frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}=\frac{2x}{2}=x$