APMO 1989 - Asian Pacific Mathematics Olympiad Question 5

Determine all functions $f:\mathbb R\to\mathbb R$ for which (1) $f(x)$ is strictly increasing, and (2) $f(x)+g(x)=2x$ for every real $x$, where $g$ is the composition inverse of $f$. (Thus $f(g(x))=x$ and $g(f(x))=x$ for every real $x$.)