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

String duality transformations in $f(R)$ gravity from Noether symmetry approach