Problem 10 - Precalculus with Calculus Previews

Question

Determine whether the given polynomial function \(f\) is even, odd, or neither even nor odd. Do not graph. \(f(x)=x^{6}-5 x^{2}+7\)

Step-by-Step Solution

Answer

The function is even because \(f(-x) = f(x)\).

Step 1: Understand the Conditions for Even and Odd Functions

A function is even if \(f(-x) = f(x)\) for all \(x\). A function is odd if \(f(-x) = -f(x)\) for all \(x\). If neither condition is met, the function is neither even nor odd.

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

Calculate \(f(-x)\) for the given function \(f(x) = x^6 - 5x^2 + 7\):\[f(-x) = (-x)^6 - 5(-x)^2 + 7 = x^6 - 5x^2 + 7\]This simplifies to \(x^6 - 5x^2 + 7\).

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

Since \(f(-x) = x^6 - 5x^2 + 7\) is the same as \(f(x)\), we have \(f(-x) = f(x)\). This satisfies the condition for \(f(x)\) being an even function.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients. The variables appear with non-negative integer exponents. Polynomials can range from the simple linear form to complex expressions like quintics or sextics.
They are fundamental in algebra and have a wide range of applications. Here are some key points to remember about polynomial functions:

• Degree: The degree of a polynomial is determined by the highest power of the variable. For example, in the polynomial function given, \(x^6\) has the highest power, so it is a sixth-degree polynomial.
• Standard Form: Polynomials are often written in descending order by degree, such as \(x^6 - 5x^2 + 7\).
• Coefficients: These are the numbers that multiply the variables. In our case, 1 is the coefficient of \(x^6\), -5 is the coefficient of \(x^2\), and 7 is a constant term.

Understanding the properties of polynomial functions is essential for identifying their behavior, including symmetry properties.

Algebraic Expressions

An algebraic expression is any mathematical phrase that includes numbers, variables, and operation symbols. It can be as simple as a single variable or include multiple terms, as we've seen in the polynomial function.
Each term in an algebraic expression operates independently but contributes to the overall function. Here's a closer look at the composition of algebraic expressions:

• Terms: An expression like our polynomial \(x^6 - 5x^2 + 7\) consists of three terms: \(x^6\), \(-5x^2\), and \(7\).
• Variables: These symbols represent numbers, usually written as letters like \(x\) in the expression.
• Constants: Numbers without variables, such as 7 in our example, are constant terms.
• Operators: Mathematics functions like addition and subtraction combine terms. The operation determines how the terms affect each other.

Grasping the basics of algebraic expressions is crucial to manipulating and simplifying polynomial functions, preparing students for further algebraic studies.

Function Symmetry

Function symmetry refers to a function's property where its graph or equation remains unchanged under certain transformations. In this context, particularly, we look at whether a function is even or odd.
Understanding these concepts helps in simplifying and predicting the behavior of functions without graphing. Here's what you need to know:

• Even Functions: These functions satisfy \(f(-x) = f(x)\). Graphically, they are symmetric about the y-axis. In the exercise's solution, we confirmed that \(f(x) = x^6 - 5x^2 + 7\) is even because \(f(-x)\) returned the same expression as \(f(x)\).
• Odd Functions: These satisfy \(f(-x) = -f(x)\). Graphically, odd functions have rotational symmetry around the origin, meaning they flip over both axes.
• Neither: If a function doesn't match the criteria for even or odd, it is neither. Such functions possess no symmetry related to the origin or axes.

Recognizing a function's symmetry can be advantageous in advanced mathematical problems, saving time and effort by avoiding unnecessary graphing.