You need to clarify:
1. The definitions of your domain/codomain. It's not clear to me what R\* is, or R\*\* for that matter.
2. The structure that the isomorphism is supposed to preserve. Is it a group isomorphism? Something else?
Classic example of a Group Isomorphism:
>f : (**R**, +) → (**R**^(>0), ×)
>
>f: x ↦ e^(x)
i.e. The exponential function is an isomorphism between the group formed by the real numbers under addition and the positive real numbers under multiplication. It is both a group homomorphism and a bijection (making it a group isomorphim).