Definition:Inverse Hyperbolic Sine/Complex

Definition
Let $\sinh: \C \to \C$ denotes the hyperbolic sine as defined on the set of complex numbers.

Let $x = \sinh y$.

Definition 2
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

 * Inverse Hyperbolic Sine Logarithmic Formulation