Question

If $f(x)=\frac{x}{x-1}$, then $\frac{(\text{ fofo........of })(x)}{19\text{ times }}$ is equal to:

• A. $\frac{x}{x-1}$
• B. ${\left(\frac{x}{x-1}\right)}^{19}$
• C. $\frac{19x}{x-1}$
• D. $x$

Answer 1

Answer: (A)

$\because f(x)=\frac{x}{x-1}$
$\therefore ($ fo $f)(x)=f\{f(x)\}=f\left(\frac{x}{x-1}\right)$
$=\frac{\frac{x}{x-1}}{\frac{x}{x-1}-1}=\frac{\frac{x}{x-1}}{\frac{x-x+1}{x-1}}=\frac{\frac{x}{x-1}}{\frac{1}{x-1}}=x.$
$\Rightarrow ($ fof of $)(x)=f(f$ of $)(x)$
$=f(x)=\frac{x}{x-1}$
$\Rightarrow \underset{19\text{ times }}{\underbrace{(f\text{ of of }\dots .0f)}}(x)=f($ fo $f)(x)$
$=f(x)=\frac{x}{x-1}$