Data structures for cplmsi bijections

Consider a continuous, piecewise linear, monotonically
strictly increasing (cplmsi) function on a real closed interval.
Clearly this determines a bijection.

The task is to determine a data structure for the cplmsi
function so that both functiion values and inverse function
values can be 'quickly' determined, using only functional
language constructs.  No imperative features allowed.

Secondary is the 'readability' of the resulting code, in
various functional languages, as this might be a matter
of indivdual taste.