Observer design for a nonlinear system in which the process dynamics equation are composed of nonlinear vector functions of scalar combinations of the states is considered. Assuming that the nonlinear functions have bounded derivatives, an observer design algorithm that requires solving just a single linear matrix inequality for exponentially convergent state estimation is developed. The developed algorithm works effectively when the involved nonlinear functions are monotonic. However, it fails when all or even some of the system functions are non-monotonic. Analytical results are presented to show that no solutions exist when all process dynamics functions are non-monotonic, no matter how small the Lipschitz constant or the Jacobian bounds of the nonlinearities. To overcome this limitation, a switched gain observer that switches between multiple constant observer gains is developed that can provide global exponentially stability for systems with non-monotonic nonlinear functions. The application of the developed hybrid observer is demonstrated to a motion estimation application involving vehicle position tracking on local roads and highways.