34.2. Find all functions $f(x)$ that are defined for $x \neq 1$ and satisfy the relation

$$
f(x) + f\left(\frac{1}{1-x}\right) = x
$$
34.2. Let $\varphi_{1}(x)=\frac{1}{1-x}$. Then $\varphi_{2}(x)=1-1 / x$ and $\varphi_{3}(x)=x$. Therefore, we obtain the system of equations

$$
\left\{\begin{array}{l}
f(x)+f\left(\frac{1}{1-x}\right)=x \\
f\left(\frac{1}{1-x}\right)+f\left(1-\frac{1}{x}\right)=\frac{1}{1-x} \\
f\left(1-\frac{1}{x}\right)+f(x)=1-\frac{1}{x}
\end{array}\right.
$$

Add the first equation to the third and subtract the second equation from them. As a result, we get

$$
f(x)=\frac{1}{2}\left(x+1-\frac{1}{x}-\frac{1}{1-x}\right)
$$

A direct check shows that this function satisfies the required relation.