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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions1 Mark
MCQ
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.
  • ✓
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • B
    Both (A) and (R) are true but (R) is not the correct explanation of (A).
  • C
    (A) is true but (R) is false.
  • D
    (A) is false but (R) is true.
✓

Answer

Correct option: A.
Both (A) and (R) are true and (R) is the correct explanation of (A).
(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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