  We study dually epi-translation invariant valuations on cones of convex
functions containing the space of finite-valued convex functions. The existence
of a homogeneous decomposition is used to associate a distribution to every
valuation of this type similar to the Goodey-Weil embedding for translation
invariant valuations on convex bodies. The relation between the valuation and
its associated distribution is used to establish a notion of support for
valuations. As an application, we show that there are no $\mathrm{SL}(n)$ or
translation invariant valuations except constant valuations in this class and
we discuss which valuations on finite-valued convex functions can be extended
to larger cones. In addition, we examine some topological properties of spaces
of valuations with compact support.
