  We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any
subset of a finite field can be decomposed into two disjoint subset $\cU$ and
$\cV$ of which the additive energy of $\cU$ and $f(\cV)$ are small, for
suitably chosen rational functions $f$. We extend the result by proving
equivalent results over multiplicative energy and the additive and
multiplicative energy hybrids.
