Given an irreducible variety and a variety a rational map (notice dashed arrow notation) is an equivalence class of partially defined maps, namely the pairs where is a regular map defined on dense Zariski open subvarieties and the equivalence is the agreement on the common intersection.
The notion of an image of a rational map is nontrivially defined, see that entry. A rational map is dominant if its image as a rational map is the whole of .
The composition of rational maps where and is not always defined, namely it is even possible that the image of lies out of any dense open subset in , where is defined as a regular map. The composition is defined as the class of equivalence of pairs where and are open dense subsets and if such exist and undefined otherwise.
If is dominant then in this situation is the composition is always defined.