Question

Determine whether the function is even, odd, or neither. $$g(x)=-2 x^6+x^2$$

Short answer

The function \( g(x) = -2x^6 + x^2 \) is even.

Step-by-step solution

Check If the Function Is Even

To determine if the function is even, replace \( x \) in the function with \( -x \) and simplify the expression. We get \( g(-x)=-2(-x)^6+(-x)^2=-2x^6+x^2 \). After simplification, it's obvious that \( g(-x) = g(x) \). Hence the function is even.

Check If the Function Is Odd

To determine if the function is odd, check whether the negative of \( g(-x) \) equals \( g(x) \). In this case, \(-g(-x) = -(g(-x)) = -(-2x^6+x^2) = 2x^6-x^2 \). After simplification it is clear that \(-g(-x) \neq g(x)\). Hence the function is not odd.

Conclude the Type of the Function

Since \( g(-x) = g(x) \), the function is even. And since \(-g(-x) \neq g(x)\), the function is not odd. Therefore, the function \( g(x) = -2x^6+x^2 \) is an even function.

Key concepts

Polynomial Functions

Polynomial functions are a fundamental concept in mathematics. They consist of terms that are made up of a constant coefficient and a variable raised to a non-negative integer power. Each term in a polynomial can be expressed as: \[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(x\) is the variable. It's important to note that the powers of \(x\) (denoted as \(n, n-1,\) etc.) are always whole numbers, which makes polynomials very versatile and widely applicable in various fields of science and engineering.
The polynomial function \(g(x) = -2x^6 + x^2\) is a classic example. It includes terms where the variable \(x\) is raised to powers of 6 and 2. This makes it a polynomial of degree 6, with \(-2\) as the coefficient of the highest power term and \(1\) as the coefficient of the \(x^2\) term. Being able to identify the degree and coefficients of polynomial functions is crucial for understanding their behavior and properties.

Function Symmetry

Function symmetry is an essential concept when analyzing and classifying functions. Understanding symmetry helps determine if a function is even, odd, or neither, which gives us insight into the structure and behavior of the function.

• **Even functions**: A function is even if it satisfies the condition \(f(-x) = f(x)\) for all \(x\). Graphically, this means the function is symmetric with respect to the y-axis.
• **Odd functions**: A function is odd if \(-f(x) = f(-x)\). This results in symmetry about the origin. If you rotate the graph 180 degrees around the origin, it looks the same.

For our function \(g(x) = -2x^6 + x^2\), we check these properties to classify its symmetry. By replacing \(x\) with \(-x\), we observed that \(g(-x) = g(x)\), indicating that the function is even. Symmetrical about the y-axis, this even property means that transformations like reflections about the y-axis will leave the function invariant.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that form the basis of equations and functions. In examining functions, especially polynomials, knowing how to manipulate these expressions is key.
For a function such as \(g(x) = -2x^6 + x^2\), understanding its composition requires familiarity with basic algebraic rules:

• **Substitution**: Replace variables with numbers or other variables to simplify or evaluate expressions.
• **Exponent rules**: Apply rules such as \((-x)^n = x^n\) for even \(n\), and \((-x)^n = -x^n\) for odd \(n\), to simplify expressions.

Using these principles, when \(x\) in \(g(x)\) is replaced with \(-x\), the expression remains the same (\(g(-x) = -2x^6 + x^2\)). This reflects the application of exponent rules and indicates the function's classification as even. Algebraic expressions help us not only solve mathematical problems but also understand the overarching structure and nature of functions.