In this paper, a novel cooperative non-orthogonal multiple access (NOMA) system is proposed, where one near user is employed as decode-and-forward relaying switching between full-duplex (FD) and half-duplex (HD) mode to help a far user. Two representative cooperative relaying scenarios are investigated insightfully. The first scenario is that no direct link exists between the base station (BS) and far user. The second scenario is that the direct link exists between the BS and far user. To characterize the performance of potential gains brought by the FD NOMA in two considered scenarios, three performance metrics outage probability, ergodic rate, and energy efficiency are discussed. More particularly, we derive new closed-form expressions for both exact and asymptotic outage probabilities as well as delay-limited throughput for two NOMA users. Based on the derived results, the diversity orders achieved by users are obtained. We confirm that the use of direct link overcomes zero diversity order of far NOMA user inherent to FD relaying. In addition, we derive new closed-form expressions for asymptotic ergodic rates. Based on these, the high signal-to-noise ratio (SNR) slopes of two users for FD NOMA are obtained. Simulation results demonstrate that: 1) the FD NOMA is superior to the HD NOMA in terms of outage probability and ergodic sum rate in the low SNR region; and 2) in delay-limited transmission mode, the FD NOMA has higher energy efficiency than the HD NOMA in the low SNR region; However, in delay-tolerant transmission mode, the system energy efficiency of the HD NOMA exceeds the FD NOMA in the high SNR region.