Stickel, Fischer, and Richert [J. Chem. Phys. 1996, 104, 2043] found the temperature dependence of the cc-relaxation time, tau(alpha)(T), of many small-molecule glass-forming liquids undergoes a change at a characteristic temperature, T-B. On the lowering of temperature, tau(alpha)(T) changes from one Vogel-Fulcher-Tammann (VFT) dependence, VFTh, that holds for T > T-B to another VFT dependence, VFT1, or a non-VFT dependence, nVFT(1), for T < T-B. In glass-formers, which exhibit a beta-relaxation of the Johari-Goldstein type, extrapolation of the Arrhenius temperature dependence of its relaxation time, tau(beta)(T), indicates that it tends to merge into the alpha-relaxation at a temperature, T-beta. Stickel et al. found also that T-B coincides with T-beta. This work augments these important findings by showing that, for fragile glass-formers, on decreasing temperature there is at T-B approximate to T-beta a rapid steplike change of the Kohlrausch-Williams-Watts exponent, (1 - n), of the primary alpha-relaxation correlation function, exp[-(t/tau(alpha))(1-n)]. The size of the steplike change of different glass-formers decreases proportionately with decreases in the value of n(T-g) at the glass transition temperature T-g and in the difference between VFTh and VFT1 or nVFT(1), after these functions are extended to a common temperature range above and below T-B. These correlations are interpreted as due to the onset of significant increase of cooperativity at and below T-B approximate to T beta as indicated by the rapidity of the increase of n(T) in the framework of the coupling model. If one compares different glass-formers ai the same value of the ratio, T-B/T, the interpretation corroborates the experimental fact that a larger steplike increase of (1 - n) across T-B gives rise to larger separation between the logarithm of the alpha and beta relaxation times. The fact that T-beta nearly coincides with T-B is also explained. For (T-B/T) > 1, the difference, log[tau(alpha)(T-B/T)] - log[tau(beta)(T-B/T)], remains small for strong glass-formers with small n(T-g) and small step increase of (1 - n) across T-B. As a result, a beta-relaxation peak or shoulder cannot be resolved from the a-relaxation peak and observed as a wing on the high-frequency side of the cr-relaxation peak. This interpretation can further explain the origin of the success of the Dixon-Nagel scaling of the dielectric loss data of glass-formers.