1. Given the function $f: \mathbf{R} \rightarrow \mathbf{R}$, for any $x, y \in$ $\mathbf{R}$, we have
$$
f(2 x+f(y))=x+y+f(x) .
$$

Find $f(x)$.
1. $f(x)=x(x \in \mathbf{R})$.

For any real number $y$, take $x=-f(y)$, we get
$$
\begin{array}{l}
f(-f(y))=-f(y)+y+f(-f(y)) \\
\Rightarrow f(y)=y(y \in \mathbf{R}) .
\end{array}
$$

Obviously, $f(x)=x$ is a solution that meets the requirements.
Therefore, the original problem has only one solution, which is
$$
f(x)=x(x \in \mathbf{R}) .
$$