Definition:Inverse Hyperbolic Sine/Real

Definition 2
From Hyperbolic Sine is Bijection over Reals and by definition 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

 * Equivalence of Definitions of Real Inverse Hyperbolic Sine


 * Real Inverse Hyperbolic Sine Function is Bijection


 * 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