Problem 20 - Precalculus : Building Concepts and Connections

Question

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$g(x)=x^{3}-3 x$$

Step-by-Step Solution

Answer

The function \(g(x) = x^{3} - 3x\) is odd. It does not fit the criteria to be considered even, nor is it neither even nor odd.

Step 1: Substitute \(-x\) for \(x\)

Substitute \(-x\) for \(x\) in the given function to get: \(g(-x) = (-x)^{3} - 3(-x)\)

Step 2: Simplify the equation

Simplify the equation to isolate \(g(-x)\), this gives: \(g(-x) = -x^{3} + 3x\)

Step 3: Compare the original and the modified function

Compare the original function \(g(x) = x^{3} - 3x\) and the modified function \(g(-x) = -x^{3}+3x\). Since \(g(-x)\) is not equal to \(g(x)\), it is not an even function. But \(g(-x)\) is equal to \(-g(x)\), indicating that it is an odd function.

Key Concepts

Understanding Odd Functions

Odd functions have a unique symmetry property, known as rotational symmetry around the origin. This means that if you were to rotate the graph of an odd function by 180 degrees around the origin, it would look the same.

Mathematically, a function \( f(x) \) is considered odd if it satisfies the condition:

• \( f(-x) = -f(x) \)

This implies that if you plug in \(-x\) into the function instead of \(x\), you should get the negative of the original function value.

For example, consider the function \( g(x) = x^3 - 3x \).
When substituting \(-x\) for \(x\), we find \( g(-x) = -x^3 + 3x \).
This result is exactly \(-g(x) = -(x^3 - 3x)\), confirming that \(g(x)\) is indeed an odd function.

Exploring Even Functions

Even functions are characterized by their mirror symmetry with respect to the y-axis. This means that for every point \((x, y)\) on the function, there is a corresponding point \((-x, y)\).

The formal definition for an even function \( f(x) \) is:

• \( f(-x) = f(x) \)

In simpler terms, replacing \(x\) with \(-x\) does not change the function's output.

Let's consider the opposite of our earlier scenario: if we had tried showing that \( g(x) = x^3 - 3x \) was even, we would need \( g(-x) = g(x) \). However, as seen, \( g(-x) = -x^3 + 3x \) is not equal to \( x^3 - 3x \). Thus, \( g(x) \) does not qualify as an even function here.

Function Classification Basics

Classifying functions based on their symmetry is an essential aspect of understanding their behavior. The two primary classifications are odd and even.

• Odd functions: These have rotational symmetry around the origin, satisfying \( f(-x) = -f(x) \).
• Even functions: These exhibit symmetry about the y-axis, with \( f(-x) = f(x) \).
• Neither: If a function does not fit the criteria for being either odd or even, it is classified as neither. This means it has no particular symmetry.

When analyzing \( g(x) = x^3 - 3x \), it is classified as an odd function due to the property \( g(-x) = -g(x) \).

Understanding these classifications helps in predicting the function's graph behavior and symmetry, facilitating more informed graph interpretations and calculations.