The amplitude death (AD) phenomenon is found in the relay system in the presence of the mixed couplings composed of attractive coupling and repulsive coupling. The generation mechanism of AD is revealed and shows that the middle oscillator achieving AD is a prerequisite to further suppress oscillation of the outermost oscillators for the paradigmatic Stuart-Landau and Rössler models. Moreover, regarding the Stuart-Landau relay system as a small motif of star network, we also observe that the mixed couplings can facilitate AD state of the whole network system. Particularly, the threshold of coupling strength is invariable with the change of network size. Our findings may shed a new insight to explore the effects of hybrid coupling on complex systems, also provide a new strategy to control dynamic behaviors in engineering science and neuroscience fields.