In (Lloyd et al. 2013), the authors write (beginning of page 3) that the quantum matrix inversion techniques presented by some of the same authors in (Harrow et al. 2008) allow to efficiently implement $e^{-ig(\rho)}$ for any simply computable function $g(x)$, using multiple copies of the density matrix $\rho$ (see (1) below for a bit more context).
Given that (Harrow et al. 2008) presents a quantum algorithm to obtain a state $|x\rangle$ proportional to $A^{-1}|b\rangle$ for a given $A$ and $|b\rangle$, it doesn't seem obvious to me how this can be used to compute $e^{-ig(\rho)}$.
To which techniques exactly are the authors referring to? And how are they applied to get the stated result?
(1) More precisely, in the (Lloyd et al. 2013) paper, before making the statement that is the object of this question, the authors describe a method to construct $e^{-i\rho t}$, to accuracy $\epsilon$, using $n$ copies of $\rho$ with $n=O(t^2 \epsilon^{-1})$.
Moreover, what is meant by simply computable function does not seem to be explained in the paper.