  We present a formalism for obtaining the statistical properties of
functionals and inverse functionals of the paths of a particle diffusing in a
one-dimensional quenched random potential. We demonstrate the implementation of
the formalism in two specific examples: (1) where the functional corresponds to
the local time spent by the particle around the origin and (2) where the
functional corresponds to the occupation time spent by the particle on the
positive side of the origin, within an observation time window of size $t$. We
compute the disorder average distributions of the local time, the inverse local
time, the occupation time and the inverse occupation time, and show that in
many cases disorder modifies the behavior drastically.
