Exercise 11, Appendix 0

Let $f$ be an increasing function, continuous and $\geq 0$, defined for $x \geq 0$, and such that $f(x) \gg e_n(x)$ for every integer $n$; show that, if $g$ is the inverse function of $f$, one has $1 \ll g(x) \ll l_n(x)$ for every integer $n$.