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).