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In this paper, we prove that the quasihyperbolic metrics are quasiinvariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of above theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping. 1991 Mathematics Subject Classification. 30C65.doi:10.1090/s0002-9947-2015-06240-0 fatcat:emdpa7enhre47dspscvzzsmdxu