It is given $f_n: \mathbb{(0,\infty)} \to \mathbb{R}$ for any positive integer $n,$
If $f_n(x) =\tan^{-1}\frac{n}{1+x(x+n)} $
I have one query $\mathbb{R}$ is $(-\infty,\infty)$, but the range of $f_n(x)$ will be $(0,\frac{\pi}{2})$ for given domain $(0,\infty)$.
Because this question is from a reputed examination, i would like to know whether the given range is correct.