\begin{definition}[Definition:Inverse Hyperbolic Cosine/Real/Definition 1]
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:
:$\forall x \in S: \map {\cosh^{-1} } x := \set {y \in \R: x = \map \cosh y}$
where $\map \cosh y$ denotes the hyperbolic cosine function.
Hence for $x > 1$, $\map {\cosh^{-1} } x$ has $2$ values.
For $x < 1$, $\map {\cosh^{-1} } x$ is not defined.
\end{definition}