2. Given the line $L: x+y-9=0$ and the circle $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, point $A$ is on line $L$, $B, C$ are two points on circle $M$, in $\triangle A B C$, $\angle B A C=45^{\circ}, A B$ passes through the center of circle $M$, then the range of the x-coordinate of point $A$ is $\qquad$
Draw the tangent line $A D$ of circle $M$ through $A$, with the tangent point being $D$, and connect $A D, M D$. If there exists a point $C$ on circle $M$, then $\angle M A D$ $\geqslant 45^{\circ} \Rightarrow \sin \angle M A D=\frac{|M D|}{|A M|} \geqslant \frac{\sqrt{2}}{2}$.
Circle $M:(x-2)^{2}+(y-2)^{2}=\frac{17}{2} \Rightarrow R=\frac{\sqrt{34}}{2}$
$\Rightarrow|A M| \leqslant \sqrt{17}$. Let $A(x, 9-x), M(2,2)$,
Then $|A M|^{2}=(x-2)^{2}+(7-x)^{2}=2 x^{2}-18 x+53 \leqslant 17 \Rightarrow x^{2}-9 x+18 \leqslant 0$ $\Rightarrow(x-3)(x-6) \leqslant 0 \Rightarrow 3 \leqslant x \leqslant 6$. Therefore, the range of the x-coordinate of point $A$ is [3,6].