A functional region is an area that focuses and is strongly supported by the main city or metropolitan area. The only way this region thrives is based on one thing. That thing could be a reservoir, a metropolitan city or something like that. A city could bring in a lot of revenue based on their economy. For example, a major city that promotes tourism could be the only thing that makes that region flourish.
The city brings in all or most of the money. A shopping mall is similar in that it also brings in money. A subway system could be the main transportation and also the main way of making money for the region. The Rocky Mountains are not a functional region.