Problem 19 - College Algebra Essentials

Question

Determine whether each function is even, odd, or neither. $g(x)=x^{2}+x$$

Step-by-Step Solution

Answer

The function \(g(x) = x^{2} + x\) is neither even nor odd.

Step 1: Identifying the Function

Firstly, recognize the function that is given. The function is \(g(x) = x^{2} + x\)

Step 2: Testing for Even Property

Substitute \(-x\) for \(x\) in \((x^{2} + x)\) to determine if the statement \(g(x) = g(-x)\) holds true. Doing this transformation, we get \(g(-x) = (-x)^{2} - x = x^{2} - x\). This is not equal to \(g(x) = x^{2} + x\), so the function is not even.

Step 3: Testing for Odd Property

Since the function didn't meet the conditions of being even, now let's check if it's an odd function by testing if the condition \(g(x) = -g(-x)\) holds true. Substituting \(-x\) for \(x\) in \((x^{2} + x)\) and multiplying by -1 yields \(-g(-x) = -((-x)^{2} - x) = -x^{2} + x\). This is not equal to \(g(x) = x^{2} + x\). Therefore, the function is not odd.

Key Concepts

Polynomial Functions

Polynomial functions are a central topic in algebra and calculus. They are expressions constructed from variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. For example, the given function \(g(x) = x^2 + x\) is a polynomial function.Key characteristics of polynomial functions include:

• They are continuous and smooth, with no breaks or sharp turns.
• They can be expressed as \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where each \(a_i\) is a coefficient, and \(n\) is a non-negative integer indicating the degree of the polynomial.
• The degree of a polynomial can give us significant information about the function's behavior, such as the number of roots and the end behavior of the graph.

Polynomial functions can have zero or more terms, and they play a crucial role in modeling various real-world phenomena.

Function Symmetry

Function symmetry is an essential property that helps in classifying functions as even, odd, or neither. This classification is based on simple transformations of the input variable.- **Even Functions**: A function \(f(x)\) is even if \(f(x) = f(-x)\) for all values of \(x\) in the domain. Graphically, they are symmetric about the y-axis. - Example: \(f(x) = x^2\) is an even function. - Even symmetry often indicates that for each point on the graph on one side of the y-axis, there is a corresponding point on the other side directly opposite.- **Odd Functions**: A function \(f(x)\) is odd if \(-f(x) = f(-x)\) for every \(x\) in the domain. Graphically, they have rotational symmetry about the origin. - Example: \(f(x) = x^3\) is an odd function. - Odd functions have the property that if \((a, b)\) is on the graph, then \((-a, -b)\) will also be.
For our function \(g(x) = x^2 + x\), testing for both even and odd properties shows that it does not meet the criteria for either. Thus, \(g(x)\) is neither even nor odd.

Algebraic Properties

Understanding the algebraic properties of a function is crucial for analyzing its behavior and graph. These properties are built upon basic operations and transformations of the function.Notable algebraic properties include:

• Substitution: It involves replacing one or more variables in a function with other expressions or numbers. In symmetry testing, we substitute \(-x\) for \(x\) to analyze symmetry.
• Addition and Subtraction: These operations allow us to combine and simplify expressions, aiding in the simplification process to determine symmetry or any transformations.
• Multiplicative Properties: Such as multiplying by -1 to test for odd symmetry, as seen when testing \(-g(-x) = -x^2 + x\).

The algebraic property examination revealed that for \(g(x) = x^2 + x\), the substitutions indicate that it doesn't fit the profiles of even or odd functions due to its mixed terms \(x^2\) and \(x\), each contributing differently to the symmetry of the function. This understanding is helpful when further exploring the distinct characteristics and graphing of the function.