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Equivalent Fractions, Exponential Implications

Date: 02/17/2011 at 14:16:08
From: Justin Robinson
Subject: x^(6/4),  What is the Domain?

Say we have this function:

   f(x) = x^(6/4)

The rational exponent here can reduce and become x^(3/2). Rewriting this
as a radical, we have SQRT(x^3). This has a domain of x being greater than
or equal to 0 to avoid a negative under the radical.

However, if you keep it as x^(6/4), the radical format is 4th-root(x^6).
If you plug in a negative here, it becomes positive due to the even power
inside the radical.

So we have the two functions with different domains. I assume the latter
does still have a domain restriction; but what would be the reasoning for
it?



Date: 02/18/2011 at 13:47:16
From: Doctor Ali
Subject: Re: x^(6/4),  What is the Domain?

Hi Justin!

Thanks for writing to Dr. Math.

When we want two functions to be equal, it is not sufficient for them only
to return the same value for different x's. They also they need to have
the same domains. So do not simplify the fractions in the exponent without
thinking carefully about the domain.

In the following case, f(x) and g(x) are surely equal, since the
operations in parentheses take precedence:

     f(x) = x^(6/4)
     g(x) = x^(3/2)

Now consider these two functions:

     f(x) = (x^6)^(1/4)
     g(x) = (x^3)^(1/2)
 
Here, f(x) and g(x) won't be equal because of my explanation earlier. They
return the same values in their common domains, but these two functions
are not equal to each other since they don't have identical domains.

Please write back if you still have any difficulties.

- Doctor Ali, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Functions

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