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/
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