Inspired by the unique geometry and novel properties of a newly proposed two-dimensional (2D) carbon allotrope called pentagraphene, we have performed first-principles calculations to study the structural stability and electronic properties of pentagermanene (pGe) modulated by chemical functionalization and biaxial tensile strain. It is observed that the 2D pGe is energetically unfavorable. However, the 2D pentagonal nanosheets can be stabilized by both hydrogenation and fluorination. Phonon dispersion spectrum and ab initio molecular dynamics simulations demonstrated that the dynamic and thermal stabilities of the two functionalized pGe nanostructures can be maintained even under a high temperature of 500 K. Our calculations revealed that both hydrogenated and fluorinated-pentagonal germanenes are semiconductors with indirect band gaps of 1.92 and 1.39 eV (2.60 and 2.09 eV by the hybrid functional), respectively. The electronic structures of the functionalized pGes can be effectively modulated by biaxial tensile strain, and an indirect to direct gap transition can be achieved for the hydrogenated pGe sheet by 6% biaxial strain. Moreover, the band gap of the hydrogenated pGe could be further tailored from 0.71 to 3.46 eV (1.16-4.35 eV by the hybrid functional) by heteroatom doping (C/Si/Sn/Pb), suggesting the semiconductor-insulator transition for differently doped nanostructures. As a result, the functionalized pGes are expected to have promising applications in nanoelectronics and nanomechanics.