APMO 1989 Problem 5

Determine all functions $f$ from the reals to the reals for which (1) $f(x)$ is strictly increasing, (2) $f(x)+g(x)=2 x$ for all real $x$, where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x))=x$ and $g(f(x))=x$ for all real x.)