MHB Plotting y=x^1/3 - Why Different Results?

[Yankel]

Dear all,

I was using the computer in order to plot the graph of

\[y=x^{\frac{1}{3}}=\sqrt[3]{x}\]

and two different plotters gave two different results. I don't understand why. Can you kindly explain ?

The results are:

View attachment 8447

View attachment 8448

Thank you !

 

[HOI]

First, of course, those are the same for x> 0 just scaled differently. As for x< 0 it looks like the first plotter is using logarithms "unthinkingly" to calculate fractional powers and the logarithm of negative numbers do not exist.

 

[I like Serena]

Generally $x^{1/3}$ is considered to be undefined for negative $x$, although some books do effectively define it as $-(-x)^{1/3}$.

Reasons are:

1. The power identity $a^{b\cdot c}=(a^b)^c$ breaks down. Consider:
   $$-1 = (-1)^{2/3\cdot 3/2} \ne ((-1)^{2/3})^{3/2} = 1^{3/2} = 1$$
2. Calculators evaluate it as $x^{0.33333}$, which is undefined for negative x.
   Note that we can only define something like $x^{1/3}$ for negative x if the power is a fraction with an odd number in the denominator, but that is generally not supported by calculators.