In the previous posts of this series, I carefully considered the definition of . Let’s now repeat this logic to consider the definition of , where is an integer. We begin with even.

A typical case is ; the graph of is shown below.

The full graph of fails the horizontal line test if is even. Therefore, we need to apply the same logic that we used earlier to define . In particular, we essentially erase the left half of the graph. By restricting the domain to , we create a new function that does satisfy the horizontal line test, so that the graph of is found by reflecting through the line .

Written in sentence form,

If is even, then means that and . In particular, this is impossible for real if .

We now turn to the case of odd. Unlike before, the full graph of (in thick blue) satisfies the horizontal line test. Therefore, there is no need to restrict the domain to define the inverse function. (shown in thin purple).

In other words,

If is odd, then means that . There is no need to give a caveat on the possible values of .

In particular, and are both undefined since there is no (real) number so that or . However, is defined and is equal to since .

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