On the basis of the first-order mean spherical approximation (FMSA) theory the behavior of Sutherland fluids with a number of parameters (gamma=3.1-36) is investigated. The investigation includes its modification by the simplified exponent approximation, renormalization group (RG) transformation, and density functional theory (DFT). For long-range parameters, the original FMSA is found sufficiently good to describe the global phase behavior, including inside the critical region. For short-range parameters, the modified FMSA by the simplified exponent approximation outside the critical region and RG transform inside the critical region are applied. For extremely short-range forces, the success is achieved by its combination with the DFT. This work gives a general sense about the capability of a theory for different ranges of potential, as well as for different thermodynamic states.