Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 1 / d / Solution

Function composition is not always commutative. The reversed function composition may be undefined when the necessary domain of a function and codomain do not match. A counterexample using functions $\mathbb{R}\to \mathbb{R}$ avoids that issue: let $f(x)=x+1$ and $g(x)=2x$. ThenBoth function compositions exist, and their values differ at every real $x$.