Graph of the Inverse Secant Function

$y=\sec^{-1}x$ has domain $\mathbb{R}\setminus(-1,1)$ and range $[0,\pi]\setminus\left\{\tfrac{\pi}2\right\}$ — that is, $\sec^{-1}:\mathbb{R}\setminus(-1,1)\to[0,\pi]\setminus\left\{\tfrac{\pi}2\right\}$.

Fig. 4.25 shows the restricted secant curve on its principal domain, and Fig. 4.26 its reflection in $y=x$: two flattening branches, one for $x\ge1$ approaching the horizontal asymptote $y=\tfrac{\pi}2$ and starting at $(1,0)$, one for $x\le-1$ approaching $y=\tfrac{\pi}2$ from the other side and reaching up to $y=\pi$, with a gap in the graph directly above $(-1,1)$. …