Understanding the Odd Root Function | Sign Preservation, Domain, Range, Symmetry, and More

Odd Root Function

The odd root function is a mathematical function that takes the nth root of a number and preserves the sign

The odd root function is a mathematical function that takes the nth root of a number and preserves the sign. It is defined for all real numbers, but has a few key properties and restrictions.

For odd values of n (such as 1, 3, 5, etc.), the odd root function is defined as:

f(x) = ∛x for n = 3
f(x) = ⁵√x for n = 5
f(x) = ⁷√x for n = 7

and so on.

Here are some properties of the odd root function:

1. Preservation of sign: The odd root function preserves the sign of the input. For example, if you take the cube root of -8, you get -2. This means that if the input is negative, the output will also be negative.

2. Domain and range: The odd root function is defined for all real numbers. However, the output will be a real number only when the input is non-negative. For example, if you take the cube root of a negative number, the result will be a complex number.

3. Symmetry: The odd root function has a symmetry about the origin. This means that if you reflect a point (x, y) across the origin, the new point will also satisfy f(x) = y.

4. Increasing function: For positive values of x, the odd root function is an increasing function. This means that as the input increases, the output also increases. For example, the cube root of 1 is 1, and the cube root of 8 is 2.

5. Continuous function: The odd root function is continuous for all real numbers. This means that if you take two values that are very close to each other, their cube roots will also be very close.

It’s important to note that the odd root function is different from even root functions, such as square root (√) or fourth root (⁴√). Even root functions do not preserve the sign of the input and are defined only for non-negative numbers.

In summary, the odd root function is a mathematical function that takes the nth root of a number while preserving the sign. It is defined for all real numbers, but the output is only real when the input is non-negative. It has properties such as sign preservation, symmetry, increasing nature, and continuity.

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