Problem 52 - College Algebra

Question

Determine whether the function is odd, even, or neither. $$h(x)=2 x-x^{3}$$

Step-by-Step Solution

Answer

The function \( h(x) = 2x - x^3 \) is odd.

Step 1: Understand Definitions

An even function is symmetrical around the y-axis, which means that \( f(-x) = f(x) \) for all \( x \). An odd function is symmetrical around the origin, which means that \( f(-x) = -f(x) \) for all \( x \). Functions that do not meet either condition are neither even nor odd.

Step 2: Compute \( h(-x) \)

To determine if the function is even or odd, calculate \( h(-x) \). Given \( h(x) = 2x - x^3 \), substitute \(-x\) for \(x\):\[h(-x) = 2(-x) - (-x)^3 = -2x + x^3\]

Step 3: Compare \( h(-x) \) with \( h(x) \) and \(-h(x)\)

Compare \( h(-x) \) with \( h(x) \) and \(-h(x)\):- \( h(x) = 2x - x^3 \)- \( h(-x) = -2x + x^3 \)- \(-h(x) = -(2x - x^3) = -2x + x^3 \)Notice, \( h(-x) = -h(x) \).

Step 4: Conclusion based on Symmetry

Since \( h(-x) = -h(x) \), the function is odd because it is symmetric about the origin.

Key Concepts

Exploring Function Symmetry

Symmetry in functions refers to the visual balance of a function's graph with respect to certain lines or points. When analyzing function symmetry, there are two main types to consider:

• Symmetry about the y-axis: This is characteristic of even functions. If you fold the graph along the y-axis, both halves would match perfectly. Algebraically, this means for a function \( f(x) \), we have \( f(-x) = f(x) \) for all x.
• Symmetry about the origin: Found in odd functions, this type of symmetry results in a graph that can be rotated 180 degrees around the origin and still look the same. For odd functions, the algebraic condition is \( f(-x) = -f(x) \).

To determine about what an individual function is symmetrical, substitute \(-x\) into the function and compare it to the original and negative versions of the function. This analysis helps decide whether a function is odd, even, or neither. Understanding symmetry is essential in graphing and interpreting functions, as it provides insights into the behavior of the function over its domain.

Function Analysis for Identifying Symmetry

Function analysis refers to the process of studying and understanding the nature of a function, including its symmetry, growth, and behavior. When specifically identifying symmetry, function analysis focuses on computing values and substitutions to compare original and modified functions.For instance, given a function \( h(x) = 2x - x^3 \), one can perform the following steps to determine its symmetry:- Compute \( h(-x) \): Substitute \(-x\) into the function. For our example, \[ h(-x) = 2(-x) - (-x)^3 = -2x + x^3. \]- Compare \( h(-x) \) to \( h(x) \): Check if \( h(-x) = h(x) \). This would imply an even function, which does not hold here.- Compare \( h(-x) \) to \(-h(x)\): Verify if \( h(-x) = -h(x) \). If true, as in this example, it confirms odd symmetry as the conditions align, proving the function to be odd.Through systematic function analysis, one effectively determines the odd or even nature of a function by leveraging these comparisons. This method involves simplicity and logic, allowing students to grasp fundamental insights into functions' properties.

Understanding Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. These types of functions are prevalent in mathematics and are expressed in the form:\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \]where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer.Key aspects of polynomial functions include:

• The degree: The highest power of \( x \) with a non-zero coefficient. It often determines the end behavior and symmetry properties of the function.
• Coefficients: These can change the shape and position of the graph. For instance, in \( h(x) = 2x - x^3 \), coefficients 2 and -1 affect the slope and direction of the graph.

Polynomials can be categorized into:- Even-degree polynomials: Sometimes symmetrical about the y-axis.- Odd-degree polynomials: Often symmetrical about the origin, which may indicate an odd function.So, understanding polynomials help in examining function properties, including why specific symmetry conditions apply to the function \( h(x) = 2x - x^3 \). This deeper insight aids students in recognizing various polynomial behaviors and their implications in mathematical contexts.