Definition:Inverse Hyperbolic Sine/Real

Definition
Let $\sinh: \R \to \R$ denote the hyperbolic sine as defined on the set of real numbers.

The inverse hyperbolic sine is a multifunction defined as:


 * $\forall x \in \R: \sinh^{-1} \left({x}\right) = \left\{{y \in \R: x = \sinh \left({y}\right)}\right\}$

From Hyperbolic Sine is Bijection over Reals and Inverse of Bijection, we have that $\sinh$ admits an inverse function over $\R$.

So from Domain of Bijection is Codomain of Inverse and Codomain of Bijection is Domain of Inverse, we have that the domain and image of hyperbolic sine over $\R$, is $\R$.

Also see

 * Definition:Real Inverse Hyperbolic Cosine
 * Definition:Real Inverse Hyperbolic Tangent
 * Definition:Real Inverse Hyperbolic Cotangent
 * Definition:Real Inverse Hyperbolic Secant
 * Definition:Real Inverse Hyperbolic Cosecant