We investigate the failure of carbon-nanotube/polymer composites by using a recently-developed hybrid quantum-mechanical/molecular-mechanical (QM/MM) approach to simulate nanotube pull-out from a cross-linked polyethene matrix. Our study focuses on the strength and failure modes of covalently-bonded nanotube-polymer interfaces based on amine, carbene and carboxyl functional groups and a [2+1] cycloaddition. We find that the choice of the functional group linking the polymer matrix to the nanotube determines the effective strength of the interface, which can be increased by up to 50% (up to the limit dictated by the strength of the polymer backbone itself) by choosing groups with higher interfacial binding energy. We rank the functional groups presented in this work based on the strength of the resulting interface and suggest broad guidelines for the rational design of nanotube functionalisation for nanotube-polymer composites.