Inverse Hyperbolic Functions and Their Graphs

Since $\sinh x$ is continuous and strictly increasing on all of $\mathbb{R}$ with range $\mathbb{R}$, it is a bijection $\mathbb{R} \to \mathbb{R}$ and therefore has a genuine inverse $\sinh^{-1} x$ defined for every real $x$, with range $\mathbb{R}$. Its graph is the mirror image of $y = \sinh x$ in the line $y=x$: it passes through the origin and rises everywhere, but much more gently than $\sinh x$ itself, flattening out for large $|x|$.

$\cosh x$ is not one-to-one on all of $\mathbb{R}$ (it is even, so $\cosh x = \cosh(-x)$), so to invert it we restrict attention to $x \geq 0$, where $\cosh x$ increases strictly from $1$ to $\infty$. On this restricted domain the inverse $\cosh^{-1} x$ is defined for $x \geq 1$ and takes values $y \geq 0$. Its graph starts at the point $(1,0)$ and rises slowly, again a reflection of the right-hand half of the catenary in $y=x$.

$\tanh x$ is a strictly increasing bijection from $\mathbb{R}$ onto the open interval $(-1,1)$, so $\tanh^{-1} x$ is defined precisely for $-1 < x < 1$, with range all of $\mathbb{R}$; as $x \to \pm 1$, $\tanh^{-1} x \to \pm\infty$, so the graph has vertical asymptotes at $x = \pm 1$ and passes through the origin, rising steeply near the edges of its domain. …