Let the function g:(- , ) (- 2 , 2 ) be given by g(u)=2 ^ -1 (e^ … - Vidyadip

MCQ

Let the function $\mathrm{g}:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be given by $g(\mathrm{u})=2 \tan ^{-1}\left(e^{\mathrm{u}}\right)-\frac{\pi}{2}$. Then, $\mathrm{g}$ is

A
even and is strictly increasing in $(0, \infty)$
B
odd and is strictly decreasing in $(-\infty, \infty)$
✓
odd and is strictly increasing in $(-\infty, \infty)$
D
neither even nor odd, but is strictly increasing in $(-\infty, \infty)$

✓

Answer

Correct option: C.

odd and is strictly increasing in $(-\infty, \infty)$

c
$ g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2} $

$ =2 \tan ^{-1} e^u-\tan ^{-1} e^u-\cot ^{-1} e^u=\tan ^{-1} e^u-\cot ^{-1} e^u $

$ g(-x)=-g(x) $

$ \Rightarrow g(x) \text { is odd } $

$ \text { and } g^{\prime}(x)>0 \Rightarrow \text { increasing. }$

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