3. Let $\mathbb{R}^{\star}$ be the set of all real numbers, except 1 . Find all functions $f: \mathbb{R}^{\star} \rightarrow \mathbb{R}$ that satisfy the functional equation
$$
x+f(x)+2 f\left(\frac{x+2009}{x-1}\right)=2010 .
$$
3. Let $g(x)=\frac{x+2009}{x-1}$. Then the given functional equation becomes
$$
x+f(x)+2 f(g(x))=2010 .
$$

Replacing $x$ with $g(x)$ in $(1 \star)$, and after noting that $g(g(x))=x$, we get
$$
g(x)+f(g(x))+2 f(x)=2010 .
$$

Eliminating $f(g(x))$ in $(1 \star)$ and $(2 \star)$, we obtain
$$
x-3 f(x)-2 g(x)=-2010 \text {. }
$$

Solving for $f(x)$ and using $g(x)=\frac{x+2009}{x-1}$, we have
$$
f(x)=\frac{x^{2}+2007 x-6028}{3(x-1)} .
$$

It is not difficult to verify that this function satisfies the given functional equation.