Question

If $f(x)=\frac{x+1}{x-1},x\ne 1$ 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,
$\left(fofof\right)\left(x\right)=f\left[f\left\{f\left(x\right)\right\}\right]=f\left(x\right)=\frac{x+1}{x-1}$
Now, $\left(fofofof\right)\left(x\right)=f\left[f\left\{f\left(x\right)\right\}\right]=f\left\{f\left(x\right)\right\}$
$=\frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}=\frac{2x}{2}=x$