What is the range of $f(x) = \tan^{-1}\left(x^{2}\right)$?

Explanation

1. 01

   $x^{2}$ takes every value in $[0, \infty)$, including 0 at $x = 0$.

2. 02

   $\tan^{-1}$ is increasing, so $f$ runs from $\tan^{-1} 0 = 0$ up towards $\frac{\pi}{2}$ without reaching it: $\left[0, \frac{\pi}{2}\right)$.