Problem 33 - Precalculus: Functions and Graphs

Question

Let \(f(x)\) be a polynomial such that the coefficient of every odd power of \(x\) is \(0 .\) Show that \(f\) is an even function.

Step-by-Step Solution

Answer

The polynomial has only even powers, making it an even function.

Step 1: Understand the Definition of an Even Function

A function \(f(x)\) is called an even function if for every \(x\) in the domain of \(f\), the equality \(f(-x) = f(x)\) holds. Our goal is to show that \(f(x)\) satisfies this condition given its polynomial form.

Step 2: Express the Polynomial Form of \(f(x)\)

Since every odd power has a coefficient of 0, \(f(x)\) can be expressed as a polynomial where all terms are even powers of \(x\). Thus, \(f(x) = a_0 + a_2x^2 + a_4x^4 + \cdots + a_{2n}x^{2n}\).

Step 3: Evaluate \(f(-x)\)

Substitute \(-x\) into \(f(x)\), yielding \(f(-x) = a_0 + a_2(-x)^2 + a_4(-x)^4 + \cdots + a_{2n}(-x)^{2n}\). Since powers are even, \((-x)^{2k} = x^{2k}\) for any integer \(k\), so \(f(-x) = a_0 + a_2x^2 + a_4x^4 + \cdots + a_{2n}x^{2n}\).

Step 4: Conclude That \(f(x) = f(-x)\)

Since \(f(-x)\) resulted in the same polynomial expression as \(f(x)\), we have \(f(-x) = f(x)\). Thus, by definition, \(f(x)\) is an even function.

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression that involves sums of powers of a variable, typically denoted as \(x\). Each term in a polynomial has a coefficient and a power of \(x\), for example, \(a_nx^n\). In the context of polynomial functions:

• The degree of the polynomial is determined by the highest power of \(x\) that is present with a non-zero coefficient.
• A polynomial with only even powers, such as \(x^2, x^4\), and so on, can significantly affect the behavior of the function.

Understanding the composition of polynomial functions is critical, since they form the foundation for various mathematical concepts, including symmetry and different functional behaviors. This exercise illustrates the special case where every odd power term is missing, leading the function to have interesting symmetric properties.

Symmetry

Symmetry in mathematics refers to a situation where a function exhibits balance or uniformity in some form. For functions, symmetry often means that the function looks the same to the left and right of a certain point, usually the origin.

For even functions, symmetry is reflective over the y-axis, meaning:

• For every point \((x, f(x))\), there is a corresponding point \((-x, f(-x))\) that has the same \(y\)-value.
• This results in the property \(f(x) = f(-x)\) for all \(x\).

This concept of symmetry is key to understanding why the polynomial function in the exercise with even powers only is labeled as an even function. Since the powers are all even, substituting \(-x\) leaves the terms unchanged, thus satisfying the symmetry condition.

Power of x

The power of \(x\) in a polynomial refers to the exponent applied to \(x\) in each term of the polynomial. Different powers contribute uniquely to the function's shape and properties:

• Odd powers, such as \(x^1, x^3\), typically introduce antisymmetry, where \(f(-x) = -f(x)\), leading to odd functions.
• Even powers, like \(x^0, x^2, x^4\), help create a symmetric shape about the y-axis due to their evenness.

When a polynomial consists only of even powers of \(x\), as seen in the exercise, each term remains unaffected by substituting \(-x\). This absence of change for negative inputs leads to the polynomial being an even function. Therefore, understanding the impact of these power rules is crucial for interpreting how polynomial structures affect overall symmetry and function behavior.