As an example, this Scientific American article describes a known area in the dog genome that metabolizes carbohydrates.
How is it that researchers are able to determine specific functions such as this?
@LanceLafontaine suggested a few ways above, but I would recommend reading this section in Alberts' Molecular Biology of the Cell, which gives a more descriptive overview of some approaches, but even that is not exhaustive.
An extremely simplistic and also not exhaustive summary:
EDIT: The link to Alberts is to the 2002 edition (which is free online; the latest edition of the textbook costs around $200), so things have come a long way since then, but the underlying approaches are still in use.