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$\newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}}\newcommand{\gt}{>}\newcommand{\amp}{&}$

The following functions all have domain $\{1,2,3,4,5\}$ and codomain $\{1,2,3\}\text{.}$ For each, determine whether it is (only) injective, (only) surjective, bijective, or neither injective nor surjective.

$f = \twoline{1 \amp 2 \amp 3 \amp 4 \amp 5}{1 \amp 2 \amp 1 \amp 2 \amp 1}\text{.}$

$f = \twoline{1 \amp 2 \amp 3 \amp 4 \amp 5}{1 \amp 2 \amp 3 \amp 1 \amp 2}\text{.}$

$f(x) = \begin{cases} x \amp \text{ if } x \le 3 \\ x-3 \amp \text{ if } x \gt 3\end{cases}\text{.}$