3.9 Given numbers $\alpha$ and $\beta$ satisfy the following two equations: $\alpha^{3}-3 \alpha^{2}+5 \alpha=1, \beta^{3}-3 \beta^{2}+5 \beta=5$, try to find $\alpha+\beta$.
[Solution] Let $f(x)=x^{3}-3 x^{2}+5 x$
$$=(x-1)^{3}+2(x-1)+3 .$$

Then the left sides of the given equations are $f(\alpha)$ and $f(\beta)$, respectively. Let
$$\begin{array}{l}
g(y)=y^{3}+2 y \\
g(\alpha-1)=f(\alpha)-3=-2 \\
g(\beta-1)=f(\beta)-3=2
\end{array}$$

Since $g(y)$ is an odd function and also an increasing function, we have
$$\alpha-1=-(\beta-1) .$$

That is, $\square$
$$\alpha+\beta=2 \text {. }$$