7. (15th "Hope Cup" Senior High School Test) The function $f(x)=\sqrt{x^{2}+2 x+2}-\sqrt{x^{2}-3 x+3}$, then the difference between the maximum and minimum values of the function $f(x)$ is $\qquad$ .
7. $f(x)=\sqrt{(x+1)^{2}+1}-\sqrt{\left(x-\frac{3}{2}\right)^{2}+\frac{3}{4}}$, can be seen as the difference in distances from a moving point $P(x$,
0) to two fixed points $A(-1,1), B\left(\frac{3}{2}, \frac{\sqrt{3}}{2}\right)$, i.e.,
$$
f(x)=|P A|-|P B| \cdot|f(x)| \leqslant|A B|=\sqrt{\left(\frac{3}{2}+1\right)^{3}+\left(\frac{\sqrt{3}}{2}-1\right)^{2}}=\sqrt{8-\sqrt{3}}
$$

Therefore, the difference between the maximum and minimum values of the function $f(x)$ is $2 \sqrt{8-\sqrt{3}}$.