\[\begin{array}{l}
a)\,\,\,y = \frac{x}{{x - \sqrt x - 6}}\\
DK:\,\,\,\left\{ \begin{array}{l}
x \ge 0\\
x - \sqrt x - 6 \ne 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x \ge 0\\
\left( {x + 2} \right)\left( {x - 3} \right) \ne 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x \ge 0\\
x \ne - 2\\
x \ne 3
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x \ge 0\\
x \ne 3
\end{array} \right.\\
\Rightarrow D = \left[ {0; + \infty } \right)\backslash \left\{ 3 \right\}.\\
b)\,\,y = \sqrt {\sqrt {{x^2} + 2x + 2} - \left( {x + 1} \right)} \\
DK:\,\,\,\sqrt {{x^2} + 2x + 2} - \left( {x + 1} \right) \ge 0\\
\Leftrightarrow \sqrt {{x^2} + 2x + 2} \ge x + 1\\
\Leftrightarrow \left[ \begin{array}{l}
x + 1 \le 0\\
\left\{ \begin{array}{l}
x + 1 \ge 0\\
{x^2} + 2x + 2 \ge {\left( {x + 1} \right)^2}
\end{array} \right.
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x \le - 1\\
\left\{ \begin{array}{l}
x \ge - 1\\
{x^2} + 2x + 2 \ge {x^2} + 2x + 1
\end{array} \right.
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x \le - 1\\
\left\{ \begin{array}{l}
x \ge - 1\\
2 > 1\,\,\,\forall x
\end{array} \right.
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x \le - 1\\
x \ge - 1
\end{array} \right. \Rightarrow x \in R.\\
\Rightarrow D = R.
\end{array}\]
