1. If a is negative, then is not a real number. So is undefined as a real number when . The same is true whenever is rational with an even denominator and odd numerator – which includes, for example, every decimal ending with an odd digit.
And if x has an odd denominator, just increasing the numerator by 1 makes the result jump across to the other side of the x-axis. So there is no natural way to “fill in” the graph, and we shall not attempt to study exponential functions with negative base.
If then is undefined, as is also for all . And at x=0 we get another undefined expression . is well defined for x>0, but since its domain is restricted it is not usually included as an “exponential function”.
If , then for all x. This is perfectly well defined but is not one-to-one (in fact it is constant) and our text does not call it an “exponential function”.