Partitions of unity and bump function

Question

Exercise $13.4$
$\quad$ Let $F:N\to M$ be a $C^\infty$ map of manifolds and $h:M\to\mathbb R$ a $C^\infty$ real-valued function. Prove that $\operatorname{supp}(F^*h)\subseteq F^{-1}(\operatorname{supp} h).$

I can not image this guestion in my mind.can you give me graph and help how ı can prove this question please.

Answer 1

It's often easier to work with the complement of support. The complement of support is the union of all open sets on which the function is identically zero. Think about the following fact:

If $h$ is identically zero on an open set $U\subset M$, then $h\circ F$ is identically zero on $F^{-1}(U)$.

Then consider taking $U=M\setminus \mathrm{supp}\, h$ here.