Example 6 If $f(x)$ is an even function and $g(x)$ is an odd function, and $f(x)+g(x)=\frac{1}{x-1}$, then $f(x)=\quad ; g(x)=$ $\qquad$ .
From the given information, we have $f(-x)+g(-x)=\frac{1}{-x-1}$, thus
$$
\left\{\begin{array}{l}
f(x)+g(x)=\frac{1}{x-1}, \\
f(x)-g(x)=-\frac{1}{x+1},
\end{array}\right.
$$

Solving these equations, we get $f(x)=\frac{1}{x^{2}-1}, g(x)=\frac{x}{x^{2}-1}$.
Solution Guidance: This problem requires solving two "unknown functions," which we refer to as a "functional equation" problem. Similar to solving equations (systems of equations) for unknown variables, we construct a "system of equations" that the "unknown functions" must satisfy, thereby solving for the "unknown functions."