How to plot the equation$x(x^2-1)^\frac{1}{3}$?

Question

I would like to plot the function $x(x^2-1)^\frac{1}{3}$. When I do it in a statistical computing package, I dont get anything between $-1$ and $1$. However, when I plot this on google, I get a complete plot and when plotting in wolfram alpha, an even different plot. I sense there is something going on with complex valued roots, but am not really sure what is going on. Can anyone tell me? Thanks!

Answer 1

Note that when $x$ is between $-1$ and $1$, that is, $|x| < 1$, then $x^2 < 1$, and so the expression under the rational exponent, $x^2 - 1$, is negative.

The problem is that there are several contradictory definitions for the principal root of negative numbers. A real-valued definition produces a negative real number from this exponent $\frac{1}{3}$; for example, in this context, $(-1)^\frac{1}{3} = -1$. But a complex-valued definition instead produces a complex number; in this context, $(-1)^\frac{1}{3} = \frac{1}{2} + \frac{i \sqrt{3}}{2}$. And so in this latter case, points in this range cannot be graphed on the real Cartesian coordinate system.

Google and your graphing calculator are using the real-valued definition. Wolfram Alpha and your statistical package use the complex-valued definition by default. This clash of definitions may be responsible for a surprisingly large proportion of the questions here on Math SE. If you're asked to produce such a graph, be sure to know exactly what context and what definitions you're working under.

See also: What are the Laws of Rational Exponents?