 Solution

ID: 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

R \to R

avoids that issue: let

f (x) = x + 1

and

g (x) = 2 x

. Then

(g \circ f) (x) = 2 x + 2, (f \circ g) (x) = 2 x + 1.

Both function compositions exist, and their values differ at every real

x

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