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\title[Math 250 Handout 13 - Inverses]{Math 250 Handout 13 - Inverses}\maketitle


 Let $f\colon A \to B$ and $g \colon B \to C$ be functions. Prove or
  disprove each of the following:
  \begin{enumerate}
  \item If $f$ and $g$ are invertible, then $gf$ is invertible..
  \item If $gf$ is invertible, then $f$ and $g$ are invertible.
  \item If $gf$ is invertible, then $f$ is invertible.
  \item If $gf$ is invertible, then $g$ is invertible.
  \item If $gf$ is an injection and $g$ is invertible, then $f$ is an
    injection.
  \end{enumerate}





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