If $f(x)=2\left(\tan ^{-1}\left(e^x\right)-\frac{\pi}{4}\right)$, then $f(x)$ is:

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

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

The correct answer is Option (3) → odd and is strictly increasing in $(-\infty, \infty)$