Graph of the Inverse Tangent Function

$y=\tan^{-1}x$ has domain the entire real line and range $\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$. Since $\tan x$ is undefined (has vertical asymptotes) at $\pm\tfrac{\pi}2$, the graph of $y=\tan^{-1}x$ lies STRICTLY between the two horizontal lines $y=-\tfrac{\pi}2$ and $y=\tfrac{\pi}2$, approaching but never touching them — these are its two horizontal asymptotes.

Fig. 4.17 shows $y=\tan x$ on $\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$, and Fig. 4.18 its reflection in $y=x$: $y=\tan^{-1}x$ on $(-\infty,\infty)$ — a flattening S-shaped curve through the origin, rising slowly as $|x|$ grows large and hugging its two horizontal asymptotes.

Notes.

(i) $\tan^{-1}x$ is strictly increasing and continuous on all of $(-\infty,\infty)$. …