GATE Civil 2020 Set 2 | GA Question: 5

Question

If $f(x) = x^2$ for each $x\in (-\infty,\infty)$, then $\large \dfrac{f\left(f\left(f(x)\right)\right)}{f(x)}$ is equal to _______.

• A. $f(x)$
• B. $(f(x))^2$
• C. $(f(x))^3$
• D. $(f(x))^4$

Answer 1

Given that, $f(x) = x^{2}.$

Now, $f\left(f\left(f(x)\right)\right) = f\left(f\left(x^{2}\right)\right) = f\left(x^{4}\right) = x^{8}$

$\therefore \dfrac{f\left(f\left(f(x)\right)\right) }{f(x)} = \dfrac{x^{8}}{x^{2}} = x^{6} = (x^{2})^{3} =  (f(x))^{3}.$

So, the correct answer is $(C).$