Question

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

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

As $x\in (e,\infty )$

$\Rightarrow \log x>1\Rightarrow \log \left(\log x\right)>\log 1$

$\Rightarrow \log \left(\log x\right)>0$

$\Rightarrow \log \left(\log \left(\log x\right)\right)>\log 0$

$\Rightarrow \log \left(\log \left(\log x\right)\right)\in (-\infty ,\infty )$

$\Rightarrow$ co-domain of $f\left(x\right)=$ Range of $f\left(x\right)$

$\Rightarrow f$ is onto

Again log function is always one to one function

$\therefore f\left(x\right)$ is both one- one & onto.

Thus $f\left(x\right)$ is invertible.