Lời giải chi tiết:

\(y = \tan \left( {\dfrac{x}{2} - \dfrac{\pi }{4}} \right) = \dfrac{{\sin \left( {\dfrac{x}{2} - \dfrac{\pi }{4}} \right)}}{{\cos \left( {\dfrac{x}{2} - \dfrac{\pi }{4}} \right)}}\)

ĐK: \(\cos \left( {\dfrac{x}{2} - \dfrac{\pi }{4}} \right) \ne 0 \Leftrightarrow \dfrac{x}{2} - \dfrac{\pi }{4} \ne \dfrac{\pi }{2} + k\pi .\)

\( \Leftrightarrow \dfrac{x}{2} \ne \dfrac{{3\pi }}{4} + k\pi . \Leftrightarrow x \ne \dfrac{{3\pi }}{2} + k2\pi \,\,\,\left( {k \in \mathbb{Z}} \right).\)

Chọn C.