Continuous tuning of material properties is highly desirable for a wide range of applications, with strain engineering being an interesting way of achieving it. The tuning range, however, is limited in conventional bulk materials that can suffer from plasticity and low fracture limit due to the presence of defects and dislocations. Atomically thin membranes such as MoS2 on the other hand exhibit high Young's modulus and fracture strength, which makes them viable candidates for modifying their properties via strain. The bandgap of MoS2 is highly strain-tunable, which results in the modulation of its electrical conductivity and manifests itself as the piezoresistive effect, whereas a piezoelectric effect was also observed in odd-layered MoS2 with broken inversion symmetry. This coupling between electrical and mechanical properties makes MoS2 a very promising material for nanoelectromechanical systems (NEMS). Here, we incorporate monolayer, bilayer, and trilayer MoS2 in a nanoelectromechanical membrane configuration. We detect strain-induced band gap tuning via electrical conductivity measurements and demonstrate the emergence of the piezoresistive effect in MoS2. Finite element method (FEM) simulations are used to quantify the band gap change and to obtain a comprehensive picture of the spatially varying bandgap profile on the membrane. The piezoresistive gauge factor is calculated to be -148 +/- 19, -224 +/- 19, and -43.5 +/- 11 for monolayer, bilayer, and trilayer MoS2, respectively, which is comparable to state-of-the-art silicon strain sensors and 2 orders of magnitude higher than in strain sensors based on suspended graphene. Controllable modulation of resistivity in 2D nanomaterials using strain-induced bandgap tuning offers a novel approach for implementing an important class of NEMS transducers, flexible and wearable electronics, tunable photovoltaics, and photodetection.