Question

Let $f:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$ be given by $f(x)=(\log (\sec x+\tan x))^3$. Then

(A) $f(x)$ is an odd function

(C) $f(x)$ is an onto function

(B) $f(x)$ is a one-one function

(D) $f(x)$ is an even function

Answer 1

SOLUTION : (i) $f(-x)=-f(x)$ so it is odd function

(ii) $\quad f^{\prime}(x)=3(\log (\sec x+\tan x))^2 \frac{1}{(\sec x+\tan x)}\left(\sec x \tan x+\sec ^2 x\right)>0$

(iii) Range of $f(x)$ is $R$ as $f\left(-\frac{\pi}{2}\right) \Rightarrow-\infty \Rightarrow f\left(\frac{\pi}{2}\right) \Rightarrow \infty$ 

Hence Option (ABC) is Correct.