Conventional exchange functionals are examined through the one-parameter progressive (OP) correlation functional from physical and numerical points of view. With an exchange functional that obeys the fundamental conditions of the exact exchange functional, the OP functional has been proved to satisfy all fundamental conditions of the exact correlation functional. In this paper, we discuss whether conventional exchange functionals satisfy these conditions or not, and propose some strict conditions for exchange functionals that are required to obey the fundamental conditions. By combining the exchange functionals with the OP correlation functional, we also evaluate chemical properties for the G2 set of molecules, and confirm that the correction for exchange functionals obviously contributes to the improvement of calculated results.