Problem 74 - College Algebra and Calculus: An Applied Approach

Question

State whether the function is even, odd, or neither. $$ g(x)=x^{3}-2 x $$

Step-by-Step Solution

Answer

The function \( g(x) = x^{3} - 2x \) is odd.

Step 1: Establishing Definition of the Function

Given function is \( g(x) = x^{3} - 2x \).

Step 2: Checking if Function is Even

Find \( g(-x) \). If \( g(-x) = g(x) \), then the function is even. Start by substituting -x into the function: \( g(-x) = (-x)^{3} -2(-x) = -x^{3} + 2x = -(x^{3} - 2x) \). So, \( g(-x) \neq g(x) \). Therefore, the function is not even.

Step 3: Checking if Function is Odd

Now, let's check if the function is odd. If \( g(-x) = -g(x) \), then the function is odd. From the computation in Step 2, we can see that \( g(-x) = -(x^{3} - 2x) = -g(x) \). This shows that the function is odd.

Key Concepts

Function Properties

Functions can have specific properties that help us understand their behavior. Two common properties are being **even** or **odd**. These properties involve how a function behaves when its input is negated.

• An **even function** is symmetric with the y-axis. If you reflect it over the y-axis, it looks the same. Mathematically, for a function to be even, it must satisfy the condition: \( f(-x) = f(x) \) for all \( x \) in the domain of the function.
• An **odd function** shows rotational symmetry about the origin. If you rotate it 180 degrees around the origin, it looks the same. Mathematically, a function is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.

The function we've examined is \( g(x) = x^3 - 2x \). By substituting \(-x\) for \(x\) in the function, we found that \( g(-x) = -(x^3 - 2x) = -g(x) \), satisfying the condition for odd functions. Therefore, \( g(x) \) is an odd function.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's a unifying thread of almost all mathematics. In our exercise, it involves checking the symmetry properties of the function \( g(x) = x^3 - 2x \).
When you see a function like this, you can follow these steps:

• Calculate \( g(-x) \) by replacing \( x \) with \( -x \). This gives insight into the function's behavior under negation.
• Determine if \( g(-x) = g(x) \), or \( g(-x) = -g(x) \). This will reveal whether the function is even or odd.

For \( g(x) \), when we compute \( g(-x) \), we get \( -x^3 + 2x \), which matches \( -g(x) = -(x^3 - 2x) \). This relationship shows us clearly that the function is odd.

Polynomial Functions

Polynomial functions are a fundamental class of functions in algebra. They're made by adding powers of \( x \) with coefficients, where the powers are non-negative integers. The function \( g(x) = x^3 - 2x \) is a polynomial. Let's break it down:

• It's a **degree 3 polynomial** because the highest power of \( x \) is 3, known as a cubic polynomial.
• This polynomial only includes odd powers of \( x \). Odd-powered polynomial terms contribute to the function's property of being odd.

Understanding properties of polynomial functions helps in predicting their general behavior, such as symmetry, which is useful in graphing and analysis. The function \( g(x) \), with both terms having odd exponents, fits perfectly into the criteria of an odd function, which matches with our findings from substituting \( -x \) into the function formula. Recognizing that functions with only odd degree terms are inherently odd functions simplifies understanding even further.