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Relations and Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRelations and Functions1 Mark
Question
Assertion (A) : Let $f:(e, \infty) \rightarrow R$ defined by $f(x)=\log (\log (\log x))$ is bijective.
Reason (R) : A function $f$ will be bijective if $f$ is both one-one and onto.
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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 logarithmic functions are always one-one.
$\therefore f(x)$ is both one - one and onto.

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