Minimum number of edges for $n$ vertex graph to be connected
Likely they mean that if a graph has as at least $f(n) = {n-1 \choose 2}+1$ edges, then it is connected. For instance, $f(5) = 7$, and any 5-vertex graph is connected with 7 edges. However, as you observe, it would also be connected with a smaller number of edges (that is, with $n-1 = 5-1 = 4$ edges).