Example 4 Let $\alpha$ and $\beta$ be the two real roots of the equation $x^{2}-22 x+m=0$ with respect to $x$. Express $|\alpha|+|\beta|$ as a function of $m$, then $|\alpha|+|\beta|=$ $\qquad$.
Solve: Since $\alpha, \beta$ are both real numbers, the discriminant $\Delta=22^{2}-4 m \geqslant 0$, i.e., $m \leqslant 121$.
If $m \geqslant 0$, then from $\left\{\begin{array}{l}\alpha+\beta=22 \\ \alpha \beta=m\end{array}\right.$ we get $|\alpha|+|\beta|=\alpha+\beta=22$;
If $m<0$, then $|\alpha|+|\beta|=|\alpha-\beta|=\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}=\sqrt{484-4 m}$. Therefore, $|\alpha|+|\beta|=\left\{\begin{array}{l}22,0 \leqslant m \leqslant 121, \\ \sqrt{484-4 m}, m<0 .\end{array}\right.$