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

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neither even nor odd, but is strictly increasing in $(- \infty, \infty)$

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

even and is strictly increasing in $(0, \infty)$

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

answer is C.

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Detailed Solution

$g (u) = 2 \tan^{- 1} (e^{u}) - \frac{π}{2} g^{'} (u) = \frac{2 e^{u}}{1 + e^{2 u}} > 0$

Hence g(u) is increasing function $g (u) = - g (u)$
Hence $g (u)$ is odd strictly increasing in $(- \propto, \propto)$

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