Definition:Real Inverse Hyperbolic Function

Definition
Let $f: \R \to \R$ be one of the hyperbolic functions on the set of real numbers.

Certain of the inverse hyperbolic function $f^{-1} \subseteq \R \times \R$ are actually multifunctions, such that for a given $y \in \R$ there may be more than one $x \in \R$ such that $y = \map f x$.

As with the inverse trigonometric functions, it is usual to restrict the codomain of the multifunction so as to allow $f^{-1}$ to be single-valued.

There are six basic hyperbolic functions, so each of these has its inverse functions:

Real Inverse Hyperbolic Cosine
Let $S$ denote the subset of the real numbers:
 * $S = \set {x \in \R: x \ge 1}$

Real Inverse Hyperbolic Tangent
Let $S$ denote the open real interval:
 * $S := \openint {-1} 1$

Real Inverse Hyperbolic Cotangent
Let $S$ denote the union of the unbounded open real intervals:
 * $S := \openint \gets {-1} \cup \openint 1 \to$

Real Inverse Hyperbolic Secant
Let $S$ denote the half-open real interval:
 * $S := \hointl 0 1$