Example 13 (2003 Hunan High School Mathematics Competition Question)
Let $x, y$, $z>0$ and $x+y+z=1$, find the minimum value of $f(x, y, z)=\frac{3 x^{2}-x}{1+x^{2}}+$ $\frac{3 y^{2}-y}{1+y^{2}}+\frac{3 z^{2}-z}{1+z^{2}}$.