Domain of x^(1/3) +x^(4/3): Why Negative Values Not Allowed?

In summary, when attempting to graph the function x^(1/3) + x^(4/3), some graphing calculators and function graphers may not plot the function for negative values of x due to how they interpret the exponent 1/3. However, there are alternative programs that can accurately graph the function.
  • #1
CalculusHelp1
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Homework Statement



Graph x^(1/3) +x^(4/3)


Homework Equations



Limits, derivatives, etc.

The Attempt at a Solution



Hi guys, I was attempting this problem and then verifying it using a function grapher online. I noticed that all of the function graphers do not plot the function for avalues of x less than 0, and my calculator also gives an error when attempting a negative number to the power of 4/3. I would assume you can raise a negative number to the power of 4/3, so why does the calculator give an error and the function graphers so that negative x is outside the domain? Any ideas?
 
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  • #2
Ah yes. Well, most of the function graphers treat 1/3 as 0.333333333... So they don't consider 1/3 as a fraction, but as a real number instead. But a negative number exponent a general real number does not have to exist. It only exists if this real number is a fraction. But function graphers fail to see that it IS a fraction. So it's really because function graphers are stupid...


I would suggest following (free) program which circumvented the problem and which would graph your function correctly: http://www.padowan.dk/graph/
 
  • #3
That's much better, thanks for the help.
 

1. Why are negative values not allowed in the domain of x^(1/3) + x^(4/3)?

Negative values are not allowed in the domain of x^(1/3) + x^(4/3) because raising a negative number to a fractional exponent results in a complex number. In other words, negative numbers do not have real cube roots or fourth roots, which are required in this expression.

2. Can't we just use imaginary numbers for the negative values?

While it is true that imaginary numbers can be used for negative values in certain situations, in this case it would not make sense to do so. The expression x^(1/3) + x^(4/3) is typically used to model real-world scenarios, and using imaginary numbers for negative values would not accurately represent those scenarios.

3. What happens if we do include negative values in the domain?

If negative values are included in the domain of x^(1/3) + x^(4/3), the resulting function would not be well-defined. This means that the function would not have a unique output for every input, which goes against the fundamental definition of a function.

4. Can we restrict the domain to only include positive values?

Yes, the domain of x^(1/3) + x^(4/3) can be restricted to only include positive values. This would result in a valid function that can be used to model real-world scenarios. However, it is important to note that this restriction may limit the applicability of the function in certain situations.

5. Is there a way to modify the expression to include negative values in the domain?

No, there is no way to modify the expression x^(1/3) + x^(4/3) to include negative values in the domain while still maintaining a valid function. The only way to include negative values would be to use a different expression or function altogether.

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