Q.Describe the graph of $y=\tan^{-1}x$, stating its domain, range and behaviour as $x\to\pm\infty$.

The graph of $y=\tan^{-1}x$ has domain all of $\mathbb{R}$ and range the open interval $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$. It is strictly increasing throughout, passes through the origin (since $\tan^{-1}0=0$), and has two horizontal asymptotes: $y\to\dfrac{\pi}{2}$ as $x\to\infty$, and $y\to-\dfrac{\pi}{2}$ as $x\to-\infty$ -- the curve gets arbitrarily close to these lines but never touches them, since the range is open. This gives it the characteristic S-shape of a bounded, monotonic function with unbounded domain.

State the domain, range, monotonicity and asymptotic behaviour, reading each off the principal branch and the reflection-in-$y=x$ construction of Section 4.

Saying the curve 'reaches' $y=\pm\pi/2$ -- the range is an open interval, so these values are only approached, never actually attained.