Explanation

(d)
The given function is
$\ \ \ \ \ \ \ \ \ \ \ f(x) = \sqrt{1 - 2x} + 2\sin^{- 1}\left( \frac{3x - 1}{2} \right)$
For domain of $f(x),\ 1 - 2x \geq 0$ and $- 1 \leq \frac{3x - 1}{2} \leq 1$
$\Rightarrow \ \ \ \ \ \ x \leq \frac{1}{2}$ and $- 2 \leq 3x - 1 \leq 2$
$\Rightarrow \ \ \ \ \ \ \ x \leq \frac{1}{2}$ and $- \frac{1}{3} \leq x \leq 1$
$\therefore$ Domain of $f(x) = \left\lbrack - \frac{1}{3},\ \frac{1}{2} \right\rbrack$