Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Is every polynomial of even degree an even function? Is every polynomial of odd degree an odd function? Explain.

Step-by-Step Solution

Answer

Not every even degree polynomial is even, nor is every odd degree polynomial odd.

Step 1: Understanding Even Functions

A function is considered even if it satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the domain of \( f \). In other words, if you replace \( x \) with \( -x \) and the function remains unchanged, then it is even. This is tested through symmetrical behavior about the y-axis.

Step 2: Understanding Odd Functions

A function is considered odd if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \). This means that the function has rotational symmetry about the origin. When you replace \( x \) with \( -x \), the result is the negative of the original function value.

Step 3: Even Degree Polynomials

A polynomial of even degree may or may not be an even function. To be an even function, each term in the polynomial must be of even degree and without any term being just a constant unless it’s included in pairs or factored from even degree terms. For example, \( f(x) = x^2+1 \) is not an even function.

Step 4: Odd Degree Polynomials

A polynomial of odd degree may or may not be an odd function. For a polynomial to be odd, all the terms must have odd degrees, and there must not be a constant term unless it can be subtracted in odd symmetry. For example, \( g(x) = x^3 - x \) is indeed an odd function, but \( h(x) = x^3 + 2 \) is not.

Key Concepts

Even Functions

An even function is a fascinating concept in algebra. If a function is even, it fulfills the condition that for every value of \( x \) in its domain, \( f(x) \) equals \( f(-x) \). This means if you plug in a negative version of any input value, the output remains the same as if you had used the positive value. This creates a mirror-like symmetry around the y-axis.

For example, imagine graphing the function \( f(x) = x^2 \). If you graphically reflect one side of the parabola along the y-axis, it matches perfectly with the other side. This symmetry is what defines an even function. However, it's important to understand that not all even-degree polynomials are even functions. Each term of the polynomial must also satisfy the even function condition, making it symmetric about the y-axis.

Odd Functions

Odd functions have their own unique properties. They satisfy the condition \( f(-x) = -f(x) \) for all inputs \( x \). This means that when you substitute \( x \) with \( -x \), the function output becomes the negative of the original output. Odd functions demonstrate rotational symmetry around the origin, not mirror symmetry.

To visualize this, consider the function \( g(x) = x^3 \). A rotation of 180 degrees around the origin will show that the function coincides with itself. This characteristic defines an odd function. Just like even-degree polynomials are not necessarily even functions, odd-degree polynomials do not guarantee that the polynomial is an odd function either. Each polynomial term specifically needs to satisfy the odd function rule to count as an odd function.

Symmetry

Symmetry in functions can either be about the y-axis or about the origin, relating directly to even and odd functions. For even functions, symmetry is about mirroring across the y-axis. Consequently, if you fold a function's graph along the y-axis and both halves match perfectly, the function most likely displays even symmetry.

Odd functions, on the other hand, demonstrate origin symmetry. Imagine spinning the function's graph around the origin by 180 degrees. If the graph looks identical after rotation, then the function is symmetric around the origin.

• Y-axis symmetry relates to even functions.
• Origin symmetry pertains to odd functions.

Understanding these symmetries can make analyzing graphs and functions more intuitive and insightful.

Degree of Polynomials

The degree of a polynomial plays a significant role in determining possible symmetry properties, but it doesn't solely decide if a polynomial is even or odd. The degree is the highest power of any term with a nonzero coefficient in a polynomial.

• An even-degree polynomial, like \( f(x) = x^2 \), doesn't automatically imply that it is an even function. Each term must comply with the even function condition.
• An odd-degree polynomial, such as \( g(x) = x^3 \), similarly doesn't guarantee that it will be an odd function. Again, each term needs to meet the odd function condition.

Ultimately, both the degree of the polynomial and the specific behavior of all its terms determine its symmetry characteristics.