Example 3. Let the function $\mathrm{f}(\mathrm{x})$ have the properties:
i) It is defined for all $x$ except $x=0$ and $x=1$;
ii) $f(x)+f\left(\frac{x-1}{x}\right)=1+x$.

Try to find the analytical expression of the function $\mathrm{f}(\mathrm{x})$.
Given, $\mathrm{f}(\mathrm{x})+\mathrm{f}\left(-\frac{\mathrm{x}-1}{\mathrm{x}}\right)=1+\mathrm{x}$. (1)
Substitute $\frac{\mathrm{x}-1}{\mathrm{x}}$ and $\frac{1}{1-\mathrm{x}}$ for $\mathrm{x}$ in (1), we get
$$
\begin{array}{l}
f\left(\frac{x-1}{x}\right)+f\left(\frac{1}{1-x}\right)=\frac{2 x-1}{x} . \\
f\left(\frac{1}{1-x}\right)+f(x)=\frac{x-2}{x-1}
\end{array}
$$

Consider (1), (2), and (3) as a system of linear equations in three variables $f(x) 、 f\left(\frac{x-1}{x}\right), f\left(\frac{1}{1-x}\right)$. Solving this system: (1) + (3) - (2) gives
$$
2 f(x)=1+x+\frac{x-2}{x-1}-\frac{2 x-1}{x} \text {, }
$$

Therefore, $f(x)=\frac{x^{3}-x^{2}-1}{2 x(x-1)}$.