Suppose $\Omega\subset R^{n}$ is an open,convex and bounded set,$f:\Omega\to\mathbb{C}$ is a smooth map.

My question:

1)when $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$? Or in order to make $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$, whether there is some non-trivial restrictions on $f$?

What's more,if we do not need $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$,then comes the following:

2)How does $f$ affect $\pi_{1}(f(\Omega))$?