Prove or disprove: $f \circ g$ is onto $\Bbb R$ then g is onto $\Bbb R$.

Question

Prove or disprove: $f \circ g$ is onto $\Bbb R$ then g is onto $\Bbb R$. I am tending towards disproving it, but I have no idea how, I saw here a few examples with discrete groups and the integers, but I have no idea. Can you please help?

Answer 1

Hint

$f:(0,\infty)\to\mathbb{R},f(x)=\ln x$

$g:\mathbb{R}\to(0,\infty),g(x)=e^x$