A Majority-Inverter Graph (MIG) is a directed acyclic graph in which every vertex represents a three-input majority operation and edges may be complemented to indicate operand inversion. MIGs have algebraic and Boolean properties that enable efficient logic optimization. They have been shown to obtain superior synthesis results as compared to state-of-the- art And-Inverter Graph (AIG) based algorithms. In this paper, we extend MIGs to Functionally Reduced MIGs (FRMIGs), analogous to the extension of AIGs to Functionally Reduced AIGs (FRAIGs). This enables the use of MIGs in a lossless synthesis design flow. We present an FRMIG based technology mapper for lookup tables (LUTs). Any MIG may be mapped to a k- LUT network. Using exact synthesis we may decompose the k- LUT network back into an equivalent MIG. We show how LUT mapping and exact k-LUT decomposition can be used to create an MIG optimization method. Finally, we present the results of applying our new optimization method and LUT mapper to both logic optimization and technology mapping.