# How to plot $y=\frac{1}{(x-4)^{1/3}}$ with mathematical softwares?

I tried to plot the function $$y=\frac{1}{(x-4)^{2/3}}$$ with Maple software. The result was:

It is easily can seen tht the graph is not correct. But if i try to plot the function $$y=\frac{1}{((x-4)^{2})^{1/3}}$$ it gives the correct graph:

ut

Now i understand how to ask plots from Maple.

But wht if i want to plot $$y=\frac{1}{(x-4)^{1/3}}$$? How to ask it from Maple?

P:S: I tried it with some other online tools such as this or this. Same problem occurs with them also.

• I have deleted my answer because I have commited a mistake. Actually, the expression $\frac{1}{(x-4)^{\frac{2}{3}}}$ is well-defined for $x < 4$. For example $(-8)^{\frac{2}{3}}=4$ in real field. The problem is that Maple (and Matlab too) takes a complex solution, so that only for $x > 4$ the curve is traced. Sorry for that! Dec 18, 2014 at 20:52

The problem is that Maple takes the complex root with less complex argument, so the cubic root of $-1$ is understood to be $e^{\frac{\pi}{3}}$. One possibility is to ask Maple to print a piecewise function, which in the $x\leqslant 4$ region should be defined as $-\frac{1}{(4-x)^{\frac{1}{3}}}$.

Indeed, the answer of the (free tool) WA to the instruction plot 1/(x-4)^(1/3) with the option applyAssumption(1,'^_Real') is as follows:

But with the option applyAssumption(1,'^_Principal'), one gets:

In the first case, the program plots the function $$x\mapsto\left\{\begin{array}{ccc}|x-4|^{-1/3}&\mathrm{if}&x\gt4\\-|x-4|^{-1/3}&\mathrm{if}&x\lt4\end{array}\right.$$ In the second case, the program plots the function $$x\mapsto\left\{\begin{array}{ccc}|x-4|^{-1/3}&\mathrm{if}&x\gt4\\\tfrac12(1-\mathrm i\sqrt3)\cdot|x-4|^{-1/3}&\mathrm{if}&x\lt4\end{array}\right.$$ which is only natural since $z=\tfrac12(1-\mathrm i\sqrt3)$ solves $z^{-3}=-1$.

I think what you look for is the surd command.

To plot $1/(x-4)^{1/3}$ in maple you can do like this:

 plot(surd((x-4),-3),x=-10..10);


The result is

In Mathematica you can do

 Plot[Surd[x - 4, -3], {x, -10, 20}]


which results in

Another way is, asking the plot of $g(x):=\frac{x-4}{|x-4|}\frac{1}{|x-4|}$ from maple. The result is:

I saw this method in the textbook Thomas Calculus 11th ed