Problem 22 - Precalculus : Building Concepts and Connections

Question

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$g(x)=3 x^{4}-2 x^{2}+1$$

Step-by-Step Solution

Answer

The function \(g(x)=3x^{4}-2x^{2}+1\) is symmetric about the y-axis and is an even function

Step 1: Replace \(x\) with \(-x\)

Substitute \(x\) with \(-x\) in the function \(g(x)=3x^{4}-2x^{2}+1\) to get \(g(-x)=3(-x)^{4}-2(-x)^{2}+1\)

Step 2: Simplify the function

Simplify the function, \(g(-x)\), to get \(g(-x)=3x^{4}-2x^{2}+1\) which is the same as the original function \(g(x)\)

Step 3: Compare the Simplified function with the original function

Since \(g(-x)\) is equivalent to \(g(x)\), the function is symmetric about the y-axis.

Step 4: Classify the function

In accordance with the definition, since the function is symmetric about the y-axis, it is an even function

Key Concepts

Even Function

A function is classified as an "even function" if it meets a specific criterion: replacing the variable \( x \) with \( -x \) yields the original function. In simpler terms, for a function \( f(x) \) to be even, we need \( f(-x) = f(x) \) for all values of \( x \).
This means that even functions are those that remain unchanged when reflected across the y-axis.

In our exercise, the function \( g(x) = 3x^4 - 2x^2 + 1 \) was evaluated. By substituting \( x \) with \( -x \), we obtain \( g(-x) = 3(-x)^4 - 2(-x)^2 + 1 \), which simplifies to \( g(-x) = 3x^4 - 2x^2 + 1 \), exactly the same as \( g(x) \).

This confirms that the function is even. Such functions are important because they reveal symmetry and can simplify calculations in mathematics and physics.

Function Classification

Classifying functions involves determining specific properties or behaviors that functions exhibit. Functions can be broadly categorized into "even," "odd," or "neither," based on their symmetries.

• **Even Functions:** As discussed earlier, these have the property \( f(-x) = f(x) \). They exhibit symmetry along the y-axis.
• **Odd Functions:** These have the property \( f(-x) = -f(x) \), which means they are symmetric about the origin. A rotation of 180 degrees around the origin leaves the function unchanged.
• **Neither:** If a function does not satisfy the conditions for being even or odd, it falls into this category.

Classification helps predict the behavior of functions without necessarily graphing them. Knowing that a function is even or odd can help in integrating, differentiating, and analyzing them further.

Y-axis Symmetry

Y-axis symmetry is a straightforward concept that indicates a function looks the same on the left and right sides of the y-axis. When a function is drawn on a graph, and folding it along the y-axis results in a complete overlap, the function is symmetric about the y-axis.

For mathematical verification, if replacing \( x \) with \( -x \) in the function results in the same expression (i.e., \( f(-x) = f(x) \)), the function has y-axis symmetry.

In the function \( g(x) = 3x^4 - 2x^2 + 1 \), the replacement \( g(-x) \) equals \( g(x) \). Therefore, this function is symmetric about the y-axis, confirming it as an even function. Recognizing y-axis symmetry can be particularly useful in simplifying the evaluation and analysis of functions.