Let $P$ be a smooth manifold and let $i : M \hookrightarrow E$ be a smooth submanifold. The subspace
of smooth functions that are transverse maps to $i$ is a dense subspace.
This is due to (Thom).
The result is due to
A review is in
John Francis, Topology of manifolds course notes (2010) (web)
Lecture 4 Transversality (notes by I. Bobkova) (pdf)