We study the effects of covalent functionalization of single wall carbon nanotubes (CNT) on their elastic properties. We consider simple organic molecules -NH, -NH2, -CH2, -CH3, -OH attached to CNTs' surface at various densities. The studies are based on the first principles calculations in the framework of density functional theory. We have determined the changes in the geometry and the elastic moduli of the functionalized CNTs as a function of the density of adsorbed molecules. It turns out that elastic moduli diminish with increasing concentration of adsorbents, however, the functionalized CNTs remain strong enough to be suitable for reinforcement of composites. The strongest effect is observed for CNTs functionalized with -CH2 radical, where the Young's modulus of the functionalized system is 30% smaller than in the pristine CNTs.