  We revisit the coupled channel $K\bar{K}$ 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 $K\bar{K}$ 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 $J/\psi\to \gamma\eta\pi^0$,
$J/\psi\to \gamma\pi^+\pi^-$ and $J/\psi\to \gamma\pi^0\pi^0$ radiative decays
through the final-state interactions. We obtain the corresponding branching
fractions $Br(J/\psi\to \gamma a_0(980) \to \gamma\eta\pi^0) = (0.47\pm0.05)
\times 10^{-7}$, $Br(J/\psi\to \gamma f_0(980) \to \gamma\pi^+\pi^-) = 0.37
\times 10^{-7} - 1.98 \times 10^{-6}$, $Br(J/\psi\to \gamma f_0(980) \to
\gamma\pi^0\pi^0) = 0.18 \times 10^{-7} - 9.92 \times 10^{-7}$, and predict
$Br(J/\psi\to \gamma a_0(980)) = 1.72 \times 10^{-8} - 3.07\times 10^{-7}$ and
$Br(J/\psi\to \gamma f_0(980)) = 1.86 \times 10^{-8} - 1.89\times 10^{-5}$.
These fractions are within the upper limits of the experimental measurements.
