In this work we present a systematic density functional theory study of the electronic properties of single-walled carbon nanotubes (SWNT) with diameters ranging from 3 to 5 Å. In this work meta-generalized-gradient approximation, hybrid, and screened exchange hybrid functionals are utilized to compute energy band gaps in these narrow SWNT. Our calculations using hybrid functionals show that the only true exceptions to the zone folding predictions are the (4,0) and (5,0) SWNT. The remaining chiral SWNT are semiconducting with band gaps that can be as large as 1.7 eV. However, the calculated energy band gaps are significantly smaller than those predicted by the zone folding scheme. This difference is primarily attributed to the σ-π hybridization present in such narrow SWNT.