Definition:Inverse Hyperbolic Cosine/Real/Definition 1

Definition
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.

Also see

 * Equivalence of Definitions of Real Inverse Hyperbolic Cosine