Let G be a graph with v vertices, e edges, and f faces. In class, we saw that v e + f = 2 for simple connected planar graphs (Euler's formula). Now, we'd like to find a more general formula that also works for graphs that are not connected. We define a connected component of G to be a non-empty set of vertices in G such that there is a path from every vertex to every other vertex, and the set is maximal (every vertex reachable from a vertex in the component is in the component). For example, the following graph has 4 faces and 3 connected components:

If G has k connected components, where k N, what is v e+f equal to? Write an expression in terms of k (and no other variables). Justify your answer briefly. (Your formula should work for all planar graphs, not just the example given above. But, doing several examples should help you notice a pattern.)