34.1. Substituting $1-x$ for $x$, we get $2 f(x)+1=(1-x) f(1-x)$. The original equation shows that $f(1-x)=\frac{x f(x)-1}{2}$. Substituting this expression into the new relation, we get $2 f(x)+1=(1-x) \frac{x f(x)-1}{2}$, which means $f(x)=\frac{x-3}{x^{2}-x+4}$. Direct verification shows that this function satisfies the required relation.