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

The identity function fixes every element, and function composition means applying the inner function first:

If $f(x)=f(y)$, applying the function $f$ gives $f(f(x))=f(f(y))$. Since $f\circ f={\mathrm{id}}_A$, this implies $x=y$, proving injectivity. Also every $y\in A$ is the image of $f(y)$, because $f(f(y))=y$, proving surjectivity.

Thus $f$ is a bijection and its inverse function is itself:Such a function is an involution.

Take the functionThen $f(f(x))=-(-x)=x$ for every real number $x$, while $f(1)=-1\ne 1$. Thus it is an involution different from the identity function.