Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 1 / d / ii / Solution

Try a linear function $f(x)=cx$. The functional equation becomes $c^{2}x=(c+1)x$ for all $x$, so $c$ must be a root of a polynomial $c^2-c-1$. Its two roots of a polynomial giveFor either choice, $c^2=c+1$ proves the required equality directly. The positive coefficient is the golden ratio.