Example 1 The minimum value of the function $f(x)=\sqrt{x^{2}+3 x+3}+\sqrt{x^{2}-3 x+3}$ is $\qquad$, where $x \in \mathbf{R}$.
Solve by transforming the function as follows:
$$
f(x)=\sqrt{\left(x+\frac{3}{2}\right)^{2}+\left(0-\frac{\sqrt{3}}{2}\right)^{2}}+\sqrt{\left(x-\frac{3}{2}\right)^{2}+\left[0-\left(-\frac{\sqrt{3}}{2}\right)\right]^{2}},
$$

The right-hand side of the equation can be seen as: the sum of the distances from a moving point \( P(x, 0) \) to two fixed points \( A\left(-\frac{3}{2}, \frac{\sqrt{3}}{2}\right), B\left(\frac{3}{2},-\frac{\sqrt{3}}{2}\right) \),
Since \( A \) and \( B \) are on opposite sides of the \( x \)-axis, we have \( f(x)=|P A|+|P B| \geqslant|A B|=2 \sqrt{3} \), and the equality can always be achieved, so the minimum value of the function \( f(x) \) is \( 2 \sqrt{3} \).