[Coda]

I think the picture makes it self-evident why it's undefined outside of that narrow domain of x values -- it simply doesn't exist there, because the original function never takes on those values. And it's constrained to that range of y values in order to make it a proper function, because it can get really hard to work with expressions that can take multiple values simultaneously.

To review:

The domain of a function is the set of all x values that the function is defined for.
The range (sometimes "codomain") of a function is the set of all y values that the function can take on.

Since an inverse function is found by switching x and y in the formula, it also follows that the domain of a function constrains the range of its inverse, and the range of a function is the domain of its inverse.