  In this work, we further study a metric modified theory of gravity which
contains a non-minimal coupling to matter, more precisely, we assume two
functions of the scalar curvature, $f_1$ and $f_2$, where the first one
generalises the Hilbert-Einstein action, while the second couples to the matter
Lagrangian. On the one hand, assuming a $\Lambda$CDM background, we calculate
analytical solutions for the functions $f_1$ and $f_2$. We consider two setups:
on the first one, we fix $f_2$ and compute $f_1$ and on the second one, we fix
$f_1$ and compute $f_2$. Moreover, we do the analysis for two different energy
density contents, a matter dominated universe and a general perfect fluid with
a constant equation of state fuelling the universe expansion. On the other
hand, we complete our study by performing a cosmographic analysis for $f_1$ and
$f_2$. We conclude that the gravitational coupling to matter can drive the
accelerated expansion of the universe.
