# Is x^fraction one-to-one or not

• intenzxboi
In summary, the function x^(fraction) is one-to-one if the power is odd, and not one-to-one if the power is even. If the denominator is even, the function is not defined for negative x. If the numerator is even, the function is not one-to-one. However, this can be canceled out if the function is xodd/odd, defined for all x and one-to-one, or xodd/even, defined only for non-negative x and one-to-one.
intenzxboi
Can anyone explain to me how x^(fraction) is a one to one or not?

i know if x raised to a even power then its not one-to-one
and x raised to a odd power is one to one

but what if the power is 4/7(even/odd) or 9/8(odd/even)

or (even/even) or (odd/odd)

If the denominator is even, the function is not defined for negative x. If the numerator is even, then the function is not one-to-one. For example, x2/3= (x1/3)2 so if x= 8, (8)2/3= 22= 4 and if x= -8, (-8)2/3= (-2)2= 4.

xodd/odd is defined for all x and is one-to-one. xeven/odd is defined for all x and is not one-to-one. xodd/even is defined only for non-negative x and is one-to-one. Since a number is even if and only if it has a factor of 2, with xeven/even you can cancel 2s until you have one of the first three cases.

so basically all we need to do it look at the numerator.. if its even then its not one to one if its odd it is one to one

HallsofIvy said:
If the denominator is even, the function is not defined for negative x. If the numerator is even, then the function is not one-to-one. For example, x2/3= (x1/3)2 so if x= 8, (8)2/3= 22= 4 and if x= -8, (-8)2/3= (-2)2= 4.

xodd/odd is defined for all x and is one-to-one. xeven/odd is defined for all x and is not one-to-one. xodd/even is defined only for non-negative x and is one-to-one. Since a number is even if and only if it has a factor of 2, with xeven/even you can cancel 2s until you have one of the first three cases.

Hmm..cancelling of 2's is a very tricky business when we confine ourselves to the reals:

Wheras $$(x^{\frac{1}{6}})^{2}$$ should have the non-negatives as its maximal domain, whereas $$x^{\frac{1}{3}}$$ has the real numbers as its maximal domain.
Thus, switching from the first expression to the second, by the mechanism of cancelling 2's, does not preserve logical equivalence..

## 1. Is there a specific method to determine if x^fraction is one-to-one or not?

Yes, there is a specific method to determine if x^fraction is one-to-one or not. To do so, you can use the horizontal line test. This test involves graphing the function and drawing a horizontal line through the graph. If the line intersects the graph at more than one point, then the function is not one-to-one. If the line only intersects the graph at one point, then the function is one-to-one.

## 2. Can x^fraction be both one-to-one and not one-to-one?

No, a function cannot be both one-to-one and not one-to-one. A one-to-one function must pass the vertical line test, meaning that every input has a unique output. If a function is not one-to-one, then there exists at least one input that has multiple outputs.

## 3. What is the difference between a one-to-one function and a many-to-one function?

The main difference between a one-to-one function and a many-to-one function is that a one-to-one function has a unique output for every input, while a many-to-one function can have multiple outputs for a single input. This means that a one-to-one function passes the vertical line test, while a many-to-one function does not.

## 4. Can x^fraction be one-to-one if the fraction is negative?

Yes, x^fraction can be one-to-one if the fraction is negative. The power of x does not determine if a function is one-to-one or not. It is the overall shape and behavior of the function that determines its one-to-one status. As long as the function passes the horizontal line test, the sign of the fraction does not matter.

## 5. Are there any real-life applications of one-to-one and many-to-one functions?

Yes, there are many real-life applications of one-to-one and many-to-one functions. One-to-one functions can be used to model relationships where each input has a unique output, such as in calculating the trajectory of a projectile. Many-to-one functions can be used to model situations where there are multiple possible outcomes for a single input, such as in a multiple-choice test or in predicting the weather. Understanding the difference between these types of functions is important in various fields such as economics, engineering, and statistics.

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