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Note that for finite domains, finding an algebraic formula that gives the output for any input is often impossible. Of course we could use a piecewise defined function, like

\begin{equation*} f(x) = \begin{cases} x+1 \amp \text{ if } x = 1 \\ x-1 \amp \text{ if } x = 2 \\ x \amp \text{ if } x = 3\end{cases}\text{.} \end{equation*}

This describes exactly the same function as above, but we can all agree is a ridiculous way of doing so.

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