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

• MHB
• Yankel
In summary, the conversation discusses the discrepancy between two plotters when graphing the equation $y=x^{\frac{1}{3}}=\sqrt[3]{x}$, with one plotter using logarithms and resulting in undefined values for negative $x$. The group also mentions that calculators evaluate $x^{1/3}$ as $x^{0.33333}$, which is undefined for negative $x$. They note that $x^{1/3}$ is generally considered undefined for negative $x$, unless the denominator of the fraction is an odd number.
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 !

#### Attachments

• plot 1.PNG
1.4 KB · Views: 79
• plot 2.PNG
2.3 KB · Views: 82
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.

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.

## 1. What is the equation y=x^1/3?

The equation y=x^1/3 represents a cubic root function, where the output (y) is the cube root of the input (x).

## 2. Why do different results occur when plotting y=x^1/3?

Different results can occur when plotting y=x^1/3 due to the nature of the function. Since the cube root function has a fractional exponent, it can produce complex outputs for negative inputs, leading to different results when graphed.

## 3. How can I graph y=x^1/3 accurately?

In order to graph y=x^1/3 accurately, it is important to plot enough points to see the overall shape of the function. This includes plotting points for both positive and negative values of x, as well as using a graphing calculator or software to generate a more precise graph.

## 4. Can I use a different exponent for x in the equation y=x^1/3?

Yes, the exponent for x in the equation y=x^1/3 can be changed to any real number. This will result in a different type of function, such as a square root function for an exponent of 1/2 or a fourth root function for an exponent of 1/4.

## 5. How is the graph of y=x^1/3 different from other polynomial functions?

The graph of y=x^1/3 is different from other polynomial functions because it has a non-integer exponent. This means that the function does not follow the traditional shape of a polynomial, and its behavior can vary for different values of x.

Replies
1
Views
1K
Replies
3
Views
1K
Replies
17
Views
2K
Replies
7
Views
1K
Replies
4
Views
2K
Replies
2
Views
3K
Replies
3
Views
1K
Replies
3
Views
1K
Replies
18
Views
2K
Replies
37
Views
3K