Composition Of A Non-Diffeomorphism And A Diffeomorphism

Question

Let $M$ and $N$ be smooth manifolds, and let $f: M \rightarrow N$ be a diffeomorphism. Suppose $g: N \rightarrow P$ is not a diffeomorphism. Is it true that $g \circ f : M \rightarrow P$ is not a diffeomorphism?

Answer 1

Yes, since $f$ is a diffeomorphism, $g=g\circ (f\circ f^{-1})=(g\circ f)\circ f^{-1}$. So, if $g\circ f$ was a diffeomorphism, $g$ would be a composition of diffeomorphisms, hence, a diffeomorphism.