3. The function
$$
y=\sqrt{12+x-x^{2}}-\sqrt{15-2 x-x^{2}}
$$

has a range of $\qquad$ .
3. $[-2 \sqrt{3}, \sqrt{6}]$.

Transform the function into
$$
y=\sqrt{(4-x)(x+3)}-\sqrt{(x+5)(3-x)} \text {. }
$$

Let $y_{1}=\sqrt{(4-x)(x+3)}, y_{2}=\sqrt{(x+5)(3-x)}$.
As shown in Figure 1, the graphs of the above two functions are semicircles
$T_{1}$ and $T_{2}$. Semicircle $T_{1}$ has a center at $\left(\frac{1}{2}, 0\right)$ and a radius of $\frac{7}{2}$; semicircle $T_{2}$ has a center at $(-1, 0)$ and a radius of 4.

Let the line $x=a$ intersect semicircles $T_{1}$ and $T_{2}$ at points $A$ and $B$, respectively. Then the directed line segment $BA$ represents the function value of the desired function when $x=a$. Clearly, when $x=-3$, $y$ attains its minimum value $-2 \sqrt{3}$; when $x=3$, $y$ attains its maximum value $\sqrt{6}$.