Functions and Their Graphs · Challenge problem

Every function on a symmetric domain splits uniquely as (even part) $+$ (odd part), namely $\tfrac12\big(g(x)+g(-x)\big)+\tfrac12\big(g(x)-g(-x)\big)$. Carry out this decomposition for $g(x)=\dfrac{1}{x-1}$ and simplify each part to a single fraction.