The structure and stability of the compounds MRg$^+$ and MRgF (Rg=Ar, Kr, and Xe; M=Co, Rh, and Ir) were investigated using the B3LYP, MP2, MP4(SDQ) and CCSD(T) methods. We reported the geometry, vibrational frequencies and thermodynamics properties of these compounds. A series of theoretical methods on the basis of wavefunction analysis, including natural bond orbitals, atoms in molecules, electron localization function, and energy decomposition analysis, were performed to explore bonding nature of the M$-$Rg and Rg$-$F bonds. These bonds are mainly noncovalent, the metal weakly interacts with Rg in MRg$^+$, but their interaction is much stronger in MRgF. The neutral molecule MRgF can be well described by the Lewis structure [MRg]$^+$F$^-$.