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

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

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

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

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