Functionalization of carbon nanotubes (CNTs) is important to increase their solubility and reactivity. This paper investigates possible ways of functionalizing CNTs using the methoxy radical and the secondary butyl anion as initiators and their reactions with ethylene and epoxide, leading to polyethylene- and polyepoxide-grafted nanotubes, with the aid of the density functional theory. The nanotubes were modeled by nanotube segments with H atoms added to the dangling bonds of the perimeter carbons. The study predicts that both free-radical and anionic functionalization methods are energetically favorable; that the resulting CNT radical and the CNT anion can react with ethylene and epoxide, respectively; and that the resulting products have free electrons and negative charges on the carbon and oxygen atoms at the free ends of ethylene and epoxide, respectively. Hence, the in situ free-radical and anionic polymerizations can propagate to produce polymer-grafted CNT. The cationic functionalization of CNTs using BF3 as an initiator was found to be infeasible.