  We select $f(R)$ gravity models that undergo scale factor duality
transformations. As a starting point, we consider the tree-level effective
gravitational action of bosonic String Theory coupled with the dilaton field.
This theory inherits the Busher's duality of its parent String Theory. Using
conformal transformations of the metric tensor, it is possible to map the
tree-level dilaton-graviton string effective action into $f(R)$ gravity,
relating the dilaton field to the Ricci scalar curvature. Furthermore, the
duality can be framed under the standard of Noether symmetries and exact
cosmological solutions are derived. Using suitable changes of variables, the
string-based $f(R)$ Lagrangians are shown in cases where the duality
transformation becomes a parity inversion.
