@Vortex I am gathering from your response that the argument q ranges over the interval [a,b].
As p and q range over the interval [a,b], the argument of f(p) ranges over [a,b], while the argument of f(p+q) ranges over [2*a,2*b]. So the domain of f is the union of those intervals. For instance, if a=10, b=12, then your equation demands that f be defined over the interval [10,12] as well as the interval [20,24].
Under that scenario, it is reasonable to expect that one prescribes the values of f on [20,24] and seeks the values of f over [10,12]. Or perhaps one prescribes the values of f over [10,12] and seeks the values of f over [20,24]. The choice depends on the application in which this equation arises.
Here I have chosen a and b on purpose so that the intervals [a,b] and [2*a,2*b] do not overlap. If they do, as in the case a=1, b=100 that you have cited as an example, the situation is more complex and the assumptions on where f is prescribed and where it needs to be calculated is not so clear. Again, the source of the application should clarify the choice.