Range of the function \( f(x)=\sqrt{\cos ^{-1}\left(\sqrt{\log _{4} x}\right)-\frac{\pi}{2}} \) ...

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Range of the function \( f(x)=\sqrt{\cos ^{-1}\left(\sqrt{\log _{4} x}\right)-\frac{\pi}{2}} \) \( +\sin ^{-1}\left(\frac{1+x^{2}}{4 x}\right) \) is
(A) \( \left(0, \frac{\pi}{2}+\sqrt{\frac{\pi}{2}}\right] \)
(B) \( \left[\frac{\pi}{2}, \frac{\pi}{2}+\sqrt{\frac{\pi}{2}}\right] \)
(C) \( \left[\frac{\pi}{6}, \frac{\pi}{2}\right] \)
(D) \( \left\{\frac{\pi}{6}\right\} \)
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