Problem 53 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=x^{4}-7 x^{2}+6$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Substitute -x into the function

To determine if the function \( f(x) = x^4 - 7x^2 + 6 \) is even or odd, substitute \(-x\) for \(x\) in the function. This gives us \( f(-x) = (-x)^4 - 7(-x)^2 + 6 \).

Step 2: Simplify \( f(-x) \)

We simplify each term of \( f(-x) \). Since \((-x)^4 = x^4\) and \((-x)^2 = x^2\), we have: \( f(-x) = x^4 - 7x^2 + 6 \).

Step 3: Compare \( f(-x) \) and \( f(x) \)

Now, compare \( f(-x) = x^4 - 7x^2 + 6 \) with the original function \( f(x) = x^4 - 7x^2 + 6 \). Both expressions are identical, meaning \( f(x) = f(-x) \).

Step 4: Determine if the function is even or odd

If \( f(x) = f(-x) \), the function is even. Conversely, if \( f(x) = -f(-x) \), it would be odd. Here, since \( f(x) = f(-x) \), the function is even.

Key Concepts

Function Analysis

When we delve into function analysis, we explore the core characteristics of a function. Function analysis involves scrutinizing a function to understand its properties and behavior.
One important aspect is determining whether a function is even, odd, or neither. This classification tells us about the function's symmetry - an integral part of function analysis. You do this by evaluating the function at

• negative x-values and comparing it with the value at positive x-values.
• There's a significant exploration into various properties such as roots, intercepts, and the behavior of the function graph.

When analyzing a function, the scientist or mathematician is, in essence, breaking down the expression to see how each part contributes to the whole picture of the graph.

Polynomial Functions

Polynomial functions are expressions involving sums of powers of the variable x, each multiplied by a coefficient. They come in different types depending on the degree or the highest power of x. For example,

• linear polynomials have the highest power of 1,
• quadratic polynomials have the highest power of 2,
• cubic polynomials have the highest power of 3, and so forth.

In our exercise, our function is a polynomial of degree 4 because the highest power is x raised to 4.
Polynomial functions are significant since they are continuous and smooth, which means that you can draw their graphs without lifting your pencil off the paper. This makes them an ideal subject for understanding basic concepts in calculus and analysis.

Symmetry in Functions

Symmetry is a fascinating aspect of functions. Understanding it helps us to simplify problems and understand the function more thoroughly. There are different types of symmetry, and one major type is symmetry about the y-axis. This is particularly relevant for even functions.
An even function satisfies the condition:

• \( f(x) = f(-x) \). This tells us that the function’s graph is mirrored across the y-axis.
• An odd function, meanwhile, adheres to the condition: \( f(x) = -f(-x) \), showing symmetry about the origin.

For our exercise, where the function \( f(x) = x^4 - 7x^2 + 6 \) was found to satisfy the condition of an even function, its graph is symmetrically mirrored around the y-axis, showing the intrinsic beauty of mathematical symmetry.