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Reconfigurable intelligent surface (RIS) has been deemed as one of potential components of future wireless communication systems because it can adaptively manipulate the wireless propagation environment with low-cost passive devices. However, due to the severe double path loss, the traditional passive RIS can provide sufficient gain only when receivers are very close to the RIS. Moreover, RIS cannot provide signal coverage for the receivers at the back side of it. To address these drawbacks in practical implementation, we introduce a novel reflection and transmission dual-functional active RIS architecture in this paper, which can simultaneously realize reflection and transmission functionalities with active signal amplification to significantly extend signal coverage and enhance the quality-of-service (QoS) of all users. The problem of joint transmit beamforming and dual-functional active RIS design is investigated in RIS-enhanced multiuser multiple-input single-output (MU-MISO) systems. Both sum-rate maximization and power minimization problems are considered. To address their non-convexity, we develop efficient iterative algorithms to decompose them into separate several design problems, which are efficiently solved by exploiting fractional programming (FP) and Riemannian-manifold optimization techniques. Simulation results demonstrate the superiority of the proposed dual-functional active RIS architecture and the effectiveness of our proposed algorithms over various benchmark schemes.