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Problem (Inverting Function Composition)

Let $f:A\to B$, $g:B\to C$, and $h:C\to D$ be bijections.

  1. Show that the inverse of $g\circ f$ is $f^{-1}\circ g^{-1}$. See the note below if you forgot how to prove something is the inverse.
  2. What is the inverse of $h\circ g\circ f$? Remember to justify your claim.

Remember that you can show a function $k:B\to A$ is the inverse of $f:A\to B$ by showing that $f\circ k=id_B$ is the identity on $B$ and that $k\circ f=id_A$ is the identity on $A$. This means you must show that $f(k(b))=b$ for every $b\in B$, and that $k(f(a))=a$ for every $a\in A$.

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