While we have made significant algorithmic developments to enable autonomous systems to perform sophisticated tasks, it remains difficult for them to perform tasks effective and safely. Most existing approaches either fail to provide any safety assurances or substantially compromise task performance for safety. In this work, we develop a framework, based on model predictive control (MPC) and Hamilton-Jacobi (HJ) reachability, to optimize task performance for autonomous systems while respecting the safety constraints. Our framework guarantees recursive feasibility for the MPC controller, and it is scalable to high-dimensional systems. We demonstrate the effectiveness of our framework with two simulation studies using a 4D Dubins Car and a 6 Dof Kuka iiwa manipulator, and the experiments show that our framework significantly improves the safety constraints satisfaction of the systems over the baselines.