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I read the answers to What are argument one can give to students on the definition $0^0$? and I noticed some people are mildly bothered that $0^x$ is discontinuous if we define $0^0=1$. However, I think is fairly obvious that if $f(x)=0^x$ then $\text{domain}(f)$ does not contain negative numbers. Surely, if $x <0$ then $$0^x = 0^{-(-x)} = \frac{1}{0^{-x}}=\frac{1}{0}$$ usually an unwelcome expression. My thought here, $0^x$ is not a proper exponential function because the natural domain of the expression $0^x$ is at largest $[0, \infty)$ and, given the convention $0^0=1$, is continuous on $(0,\infty)$. In other words, it is the constant function $g(x)=0$ restricted to $(0,\infty)$. On the other hand, exponential functions with $a>0$ and $a \neq 1$ all have nice properties: they're injective functions with $\text{range}(a^x) = (0, \infty)$ and $\text{domain}(a^x) = (-\infty, \infty)$. Of course, $a=1$ is somewhat boring $a^x=1^x=1$ hence we just have the constant function with an awkward formula.

Ok, enough about the happy place. What then about $f(x)=a^x$ where $a<0$. What is the natural domain for this formula? This leads to my question:

Do you mention the problem of understanding $f(x)=a^x$ for $a<0$ in your teaching?
Is it wise to omit discussion of $a \leq 0$ ?

I don't have a firm opinion on this, I usually find myself talking about the bizarre behavior of such a formula sometime in the first week of calculus. A secondary question, does appreciating the weirdness of negative base "exponential functions" help explain why we shouldn't have much hope for $0^x$ to behave nicely? I put quotes here because I think the definition of an exponential function properly assumes the base is a positive number which is not one.

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  • $\begingroup$ In Germany, this is part of secondary education, not undergraduate education. $\endgroup$
    – Toscho
    Jun 20, 2014 at 10:19
  • $\begingroup$ A proper answer would depend on how you define the function $b^x$ where $b \in \mathbb{R}_+$ and $x\in \mathbb{R}$. (a) you can define it by first defining it for $x\in \mathbb{N}$, followed by $x\in \mathbb{Q}$, and then extending to $\mathbb{R}$ by continuity. (b) You can also "define" $b^x := \exp( \ln(b) x)$. The case $b < 0$ in case (a) requires finding (among other things) $b^{-1/2n}$ which leads naturally to complex numbers. In case (b) you need to consider $\ln(b)$ for $b < 0$... which you can of course also deal with complex analysis if you allow multiple-valued functions. $\endgroup$
    – Willie Wong
    Jun 20, 2014 at 12:34
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    $\begingroup$ My point is that some avenues of approach naturally leads to the discussion of branch cuts in complex analysis, which is a useful but deep subject. If you feel comfortable going the whole way: sure, go ahead. But if you don't feel comfortable showing them the whole story, maybe it is easier to just not confuse them needlessly. $\endgroup$
    – Willie Wong
    Jun 20, 2014 at 12:35
  • $\begingroup$ @Toscho I added the secondary education tag in view of the international nature of the MESE :) $\endgroup$
    – James S. Cook
    Jun 21, 2014 at 1:46

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I think I do the same as what you described, but in pre-calc. I try to get the students to tell me what different bases would do. I show them how bizarre a graph of (-2)^x would be, and then we dismiss negative numbers as possible bases.

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When you cover exponential functions, you should always discuss negative bases and base 0. But using power laws, it's relatively easy to see, that $(-2)^x$ is problematic for non-integer $x$, and $0^x$ is problematic for negative $x$. If you use permanence sequences, you can also easily see, that $0^x$ is also problematic for $x=0$.

These discussions are an important repetition of power laws and an important step in understanding functions (and sequences) in general. So no, it is not wise to omit non-positive bases.

And no, the weirdness of negative bases is no indicator for problems with 0. Root functions behave perfectly fine for radicand 0, only negative radicands are problematic. Exponential functions are applications of multiplication and the 0 (not negative numbers) are very special to multiplication.

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  • $\begingroup$ thanks for your answer, however, I must say about your link, Ich verstehe nicht ;) $\endgroup$
    – James S. Cook
    Jun 21, 2014 at 1:51
  • $\begingroup$ @JamesS.Cook I understand, I just didn't find English sources for it. This seems to be a traditionally German part of math didactics. $\endgroup$
    – Toscho
    Jun 21, 2014 at 9:20

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