Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 1 / f

The function composition of two bijections is a bijection. Its inverse function is two-sided, soThis is exactly the defining cancellation property of an inverse function, with the whole function composition regarded as one function.

The inverse factors are generally in the wrong order. For a counterexample, take the bijections $f(x)=x+1$ and $g(x)=2x$, with inverse functions $f^{-1}(x)=x-1$ and $g^{-1}(x)=x/2$. ThenThusFor particular bijections that commute the equality can hold; the claim fails as a universal assertion.

Use the associativity of function composition to cancel adjacent inverse functions in the right order:Therefore

For the bijections $f(x)=x+1$ and $g(x)=2x$ used above, $f\circ g$ sends $x$ to $2x+1$. Its inverse function sends $x$ to $(x-1)/2$, whereas $f^{-1}\circ g^{-1}$ sends $x$ to $x/2-1$. Hence

The proposed inverse function cancels on both sides:The uniqueness of an inverse function therefore givesThe operation applied last must be undone first.