Question

Assertion (A) : Let $f : (e, \infty) \to R$ defined by $f(x) = log (log \,(log\,x))$ is invertible.
Reason (R) : $ f$ is both one-one & onto.

• A. Both (A) & (R) are individually true & (R) is correct 3. explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (A)

As $x \in (e, \infty)$
$\Rightarrow log\,x > 1$
$\Rightarrow log (log \,x) > log \,1$
$\Rightarrow log (log \,x) > 0$
$\Rightarrow log (log (log \,x)) > log \,0$
$\Rightarrow log (log (log \,x)) \in (-\infty, \infty)$
$\Rightarrow $ codomain of $f(x) =$ Range of $f(x)$
$\Rightarrow f$ is onto
Again log function is always $ 1 - 1$
$\therefore f(x)$ is both one - one & onto.