We revisit the coupled channel $Kbar{K}$ interactions and dynamically generate the resonances $f_0(980)$ and $a_0(980)$ within both the isospin and the physical bases. The $f_0(980)-a_0(980)$ mixing effects are generated in the scattering amplitudes of the coupled channels with the physical basis, which exploits the important role of the $Kbar{K}$ channel in the dynamical nature of these resonances. With the scattering amplitudes obtained, we investigate the $f_0(980)$ and $a_0(980)$ contributions to the $J/psito gammaetapi^0$, $J/psito gammapi^+pi^-$ and $J/psito gammapi^0pi^0$ radiative decays through the final-state interactions. We obtain the corresponding branching fractions $Br(J/psito gamma a_0(980) to gammaetapi^0) = (0.47pm0.05) times 10^{-7}$, $Br(J/psito gamma f_0(980) to gammapi^+pi^-) = 0.37 times 10^{-7} - 1.98 times 10^{-6}$, $Br(J/psito gamma f_0(980) to gammapi^0pi^0) = 0.18 times 10^{-7} - 9.92 times 10^{-7}$, and predict $Br(J/psito gamma a_0(980)) = 1.72 times 10^{-8} - 3.07times 10^{-7}$ and $Br(J/psito gamma f_0(980)) = 1.86 times 10^{-8} - 1.89times 10^{-5}$. These fractions are within the upper limits of the experimental measurements.
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