\begin{definition}[Definition:Inverse Hyperbolic Cosine/Real/Definition 2]
Let $S$ denote the subset of the real numbers:
:$S = \set {x \in \R: x \ge 1}$
The '''inverse hyperbolic cosine''' $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:
:$\forall x \in S: \map {\cosh^{-1} } x := \map \ln {x \pm \sqrt {x^2 - 1} }$
where:
:$\ln$ denotes the natural logarithm of a (strictly positive) real number.
:$\sqrt {x^2 - 1}$ denotes the square root of $x^2 - 1$
Hence for $x > 1$, $\map {\cosh^{-1} } x$ has $2$ values.
For $x < 1$, $\map {\cosh^{-1} } x$ is not defined.
\end{definition}