Solving a functional equation $f(x)+f\left(\frac{1}{1-x}\right)=x$ I was given the following homework:
list all functions $f:\mathbb{R}\setminus\{0,1\}\rightarrow\mathbb{R}$ such that $f(x)+f\left(\frac{1}{1-x}\right)=x$. And obviously have I no idea what should I do here. A step-by-step explanation welcome. 

The system matrix is 
$\left( {\begin{array}{*{20}{c}}
1&1&0&& x\\
0&1&1&& {\frac{1}{{1 - x}}}\\
1&0&1&& {\frac{{x - 1}}{x}}
\end{array}} \right)$
The solution of the system is 
$\left\{ {\frac{{1 - x + {x^3}}}{{2\left( { - 1 + x} \right)x}},\frac{{ - 1 + x - 2{x^2} + {x^3}}}{{2\left( { - 1 + x} \right)x}},\frac{1}{2}\left( {1 + \frac{1}{{1 - x}} - \frac{1}{x} - x} \right)} \right\}$
so $f(x)={\frac{{1 - x + {x^3}}}{{2\left( { - 1 + x} \right)x}}}$
