Problem - 2449
Let $f(x)$ be an odd function and $g(x)$ be an even function. If $f(x)+g(x)=\frac{1}{x-1}$, find $f(x)$ and $g(x)$.
Because $f(-x)=-f(x)$ and $g(-x)=g(x)$, we have
$$-f(x)+g(x)=\frac{1}{-x-1}$$
Solving these two relations yields:
$$f(x) = \frac{x}{x^2-1}\quad\text{and}\quad g(x)=\frac{1}{x^2-1}$$