Monotone functions analyze

Question

Let $f,g : \mathbb R \to\mathbb R$. Suppose that $f\circ g$ is a strictly monotone growing function and $f$ is a strictly monotone decreasing function.

Can I conclude from those details that $g$ is a strictly monotone decreasing function?

Answer 1

Without adding "strictly" in front of "monotone", then no. $f(x) = 1$ is both monotone increasing and monotone decreasing, and no matter what $g$ is, the same will apply to $f\circ g$.