Polynomial-time Computable $f \circ g$. What does this implies for $f$ and $g$

Question

Suppose that $f$ and $g$ are functions and $f \circ g$ is polynomial computable.

a) is it true that $f$ is also polynomial computable

b) is it true that $g$ is also polynomial computable

c) if we replace "polynomial" by "logarithmic" would the previous questions be true?

Answer 1

What if $f = 0$?

What if $g = 0$ and $f(0) = 0$?