How exactly is the function $x^a$ defined

calculusexponentiationreal-analysis

I need some refresher here since the more I think about this the more I get confused.

How exactly is the function $f(x) = x^a$ defined (a – real constant, $x$ – real variable)?

1)

What is its domain? E.g. if $a=1/3$ then I know we can plug any values of $x$ so it seems the domain is the full set of all reals in this case. Is this correct?

But if we have e.g. $a=1/4$ then only non-negative values for $x$ are allowed.

Also… to prove that the function $x^a$ is continuous in real analysis they often use the representation

$$x^a = b^{a \cdot log_b{x}}$$ for some $b > 0, b \neq 0$

and then refer to the fact that the functions $b^x$ and $log_b{x}$ and the constant function $a$ are all continuous. But if so… this proof of continuity is incomplete, because for negative values of $x$ we cannot use this representation.

Or maybe… is it the case that even when $a=1/3$ (in real analysis) we still assume $x$ has to be positive? I think my real analysis (university) book assumes just that but then… I remember from high-school that I can take cubic roots of negatives so… Where is the catch?

2)

In relation to 1)… how are we supposed to understand this expression/function $x^{\sqrt3}$? Does that expression require $x$ to be positive in order to have any meaning?

I mean, for negative $x$ there is a similar problem here. We cannot just approximate $\sqrt3$ with ANY converging sequence of rationals $\frac{p_k}{q_k}$ (for k=1,2,3,…) and then define that $x^{\sqrt3}$ is the limit of the corresponding sequence:

$$x^\frac{p_1}{q_1}, x^\frac{p_2}{q_2}, …, x^\frac{p_n}{q_n}, …$$

Why? Because some of these rationals $\frac{p_i}{q_i}$ may have even denominators and then that means $x^\frac{p_i}{q_i}$ is not defined (when x is negative).

3)

All these thoughts bring me to the ultimate confusion.

What is really the co-domain of $g(x) = x^{\frac{2}{6}}$ (or say what is its value for $x=-27$)?

Is this function equal to $\sqrt[6]{x^2}$ and thus generating only positive values no matter what sign $x$ has… or is it $x^{\frac{1}{3}}$ i.e. $\sqrt[3]{x}$ and thus generating both positive and negative values?

Best Answer

Common convention is

  • no problem with positive basis (and the log representation is fine);

  • no problem with zero basis and positive exponent;

  • $0^0$ can be $0$ or $1$ depending on contexts; negative power not allowed;

  • if the basis is negative,

    • a rational exponent must be written in simplified form and have an odd denominator. The rule $x^{m/n}=\sqrt[n]{x^m}=(\sqrt[n]x)^m$ works.

    • an irrational exponent is not allowed.

If you allow complex answers, then

  • rational powers define $n$ distinct branches,

  • irrational powers define a principal branch $\sqrt[m/n]{-x}\text{ cis}(\frac{m\pi}n)$ or are not allowed.

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