Penta-graphene is a new two-dimensional metastable carbon allotrope composed entirely of carbon pentagons with unique electronic and mechanical properties. In this work, molecular dynamics simulations are carried out to investigate the effects of functionalization by hydrogen, epoxide or hydroxyl groups on the mechanical properties and failure mechanism of penta-graphene, as well as the effects of different functionalization coverages. The effects of functionalization on the structural transformation of free-standing penta-graphene triggered by increasing temperature have also been studied. The results indicate that each of the three functional groups considered can effectively tune the mechanical properties and the failure mechanism of penta-graphene. Both the Young's modulus and elastic limit of penta-graphene first decrease sharply and then increase slowly with the increase of the functionalization coverage, while the ultimate elastic strain increases monotonically. Like the pristine penta-graphene, partially functionalized penta-graphene still exhibits a plastic deformation failure behaviour under tensile load, which is caused by the irreversible pentagon-to-polygon structural transformation occurring during tensile loading. Temperature can trigger structural reconstruction for free-standing partially functionalized penta-graphene, and the corresponding critical transition temperature is higher than that of pristine penta-graphene. However, complete functionalization can change the deformation mechanism of penta-graphene from plastic deformation to brittle fracture. For fully functionalized penta-graphene by each of the three functional groups, the structural transformation is not observed when tensile strain is applied or environmental temperature is increased. These findings are expected to provide important guidelines for effectively tuning the mechanical properties of two-dimensional nanomaterials including penta-graphene.