Question

Determine if the function \(g(x)=\frac{\sqrt[3]{x}}{x^{3}-x}\) is even, odd, or neither.

Short answer

The function \(g\(x\)\) is odd.

Step-by-step solution

Identify the function

The function given is \[g(x)=\frac{\sqrt[3]{x}}{x^{3}-x}\]

Check if the function is even

A function is even if for every x in the function’s domain, \[g(-x) = g(x).\] Substitute \(-x\) into the function: \[g(-x) = \frac{\sqrt[3]{-x}}{(-x)^3 - (-x)} = \frac{-\sqrt[3]{x}}{-x^3 + x} = \frac{-\sqrt[3]{x}}{-(x^3 - x)} = \frac{-\sqrt[3]{x}}{-1 * (x^3 - x)} = \frac{-\sqrt[3]{x}}{-1} * \frac{1}{x^3 - x} = \sqrt[3]{x}\frac{1}{x^3 - x} = \sqrt[3]{x}\frac{1}{x^3 - x} \]Thus, \(g(-x) = \frac{\sqrt[3]{x}}{x^3 - x} \).If \(g(-x)\) equals \(g(x)\), the function is even. In this case, \(g(-x)\) is NOT equal to \(g(x)\), so \(g(x)\) is not even.

Check if the function is odd

A function is odd if for every x in the function’s domain, \(g(-x) = -g(x)\). Substitute \(-x\) into the function: \[g(-x) = \frac{\sqrt[3]{-x}}{(-x)^3 - (-x)} = \sqrt[3]{x} \frac{1}{x^3 - x}\] Compare \(-g(x)\): \[g(-x) = -g(x)\] It turns out that \(g(-x)\) is equal to \(-g(x)\), so it is odd.

Conclusion

Since \({g(-x) = -g(x)}\), the function \(g\(x\)\) is odd.

Key concepts

function properties

Understanding the properties of functions is crucial in math. A function maps inputs to outputs following a specific rule. To better analyze a function, we often check its domain and range.
The domain is the set of all possible input values, and the range is the set of possible output values.
Additionally, we classify functions based on their behavior, such as even or odd functions.
Knowing these properties helps in solving algebraic and trigonometric problems more effectively.

algebraic functions

Algebraic functions are created using algebraic expressions. They include polynomial, rational, and radical functions.
For example, the given function which is \( g(x)=\frac{ \sqrt[3]{x} }{x^{3}-x}\) is a type of rational function because it involves a ratio of two polynomials.
Analyzing algebraic functions involves understanding their characteristics like asymptotes, intercepts, and how they behave under various transformations.

symmetry in functions

Symmetry plays an important role in classifying even and odd functions.
A function is considered even if it is symmetric about the y-axis. This means \(f(-x) = f(x)\) for all x within the function's domain.
Conversely, a function is odd if it is symmetric about the origin, meaning \(f(-x) = -f(x)\).
In this exercise, we determined that the function \( g(x)=\frac{ \sqrt[3]{x} }{x^{3}-x}\) is odd because it satisfies \( g(-x) = -g(x) \).

function transformations

Transformations help us understand how functions change. These changes include translations, stretches, compressions, and reflections.
For instance, the transformation of replacing x with -x in the function helps detect symmetry.
In this exercise, we substituted -x into the original function and compared the result.
Detecting such transformations is essential in graphing and simplifying complex function problems.