Problem 4 - Advanced Engineering Mathematics

Question

Determine whether the function is even, odd, or neither. $$ f(x)=x^{3}-4 x $$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Definition of Even and Odd Functions

A function \( f(x) \) is said to be even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). It is said to be odd if \( f(-x) = -f(x) \) for all \( x \). Otherwise, the function is neither even nor odd.

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

To determine if a function is even or odd, first calculate \( f(-x) \). For \( f(x) = x^3 - 4x \), substitute \(-x\) into the equation:\[f(-x) = (-x)^3 - 4(-x) = -x^3 + 4x.\]

Step 3: Compare \( f(-x) \, \text{with} \, f(x) \)

Now, compare \( f(-x) = -x^3 + 4x \) to \( f(x) = x^3 - 4x \).- \( f(-x) \) is not equal to \( f(x) \), so the function is not even.- Check if \( f(-x) = -f(x) \): \[ -f(x) = -(x^3 - 4x) = -x^3 + 4x \] Since \( f(-x) = -f(x) \), the function is odd.

Step 4: Conclude the Nature of the Function

Based on the comparison steps, since \( f(-x) = -f(x) \), the function \( f(x) = x^3 - 4x \) is odd.

Key Concepts

even functions

Even functions have a characteristic property where the function's value remains the same even if you flip the input. Mathematically, a function \( f(x) \) is said to be even if \( f(-x) = f(x) \) for every \( x \) within its domain. This means that the graph of an even function is symmetric about the y-axis. In simpler terms, if you were to fold the graph over the y-axis, both halves would match perfectly. Examples of even functions include:

• The quadratic function \( f(x) = x^2 \)
• The cosine function \( f(x) = \cos{x} \)

Studying even functions is essential as they help us understand the behavior of graphs and provides insights into their symmetry properties. Even though the function we examined, \( f(x) = x^3 - 4x \), is not even, knowing how to identify even functions is a crucial part of mathematical analysis.

function comparison

Function comparison is a method used to determine the nature of a function by evaluating how it behaves when the inputs are altered. The process starts by determining \( f(-x) \) and comparing it to the original function \( f(x) \). There are three possibilities:

• If \( f(-x) = f(x) \), the function is even.
• If \( f(-x) = -f(x) \), the function is odd.
• If neither condition is met, the function is neither even nor odd.

In our specific case, when we evaluated \( f(x) = x^3 - 4x \), comparing \( f(-x) = -x^3 + 4x \) with \( f(x) \)follows the second scenario, indicating that the function is odd. This comparison not only categorizes the function but also unveils crucial symmetries and behavior in its graph.

function symmetry

Functions can have different types of symmetry which make them easier to analyze graphically. Symmetry in functions refers to maintaining some form of balance or replicable patterning as you alter inputs or reflections, usually about the y-axis or the origin.

• Y-axis Symmetry: Even functions exhibit y-axis symmetry, meaning their graph is mirrored over the y-axis.
• Origin Symmetry: Odd functions display origin symmetry. When you rotate the graph 180 degrees around the origin, it matches the original. For the function \( f(x) = x^3 - 4x \), since it is odd, it will present this origin symmetry.

Recognizing these symmetry patterns simplifies complex function analysis and can facilitate solving equations, integrals, and various mathematical analyses.

mathematical analysis

Mathematical analysis is a branch of mathematics that deals with limits, functions, derivatives, integrals, and infinite series. One foundational aspect of analysis is examining the properties of functions, like determining if they are even or odd. This kind of analysis allows us to:

• Predict the behavior of functions.
• Understand functional properties and their implications in larger systems.
• Perform calculus operations more efficiently by exploiting symmetrical properties.

In this particular case, recognizing that \( f(x) = x^3 - 4x \) is an odd function helps predict its overall behavior. Mathematical analysis is crucial in modeling and solving real-world problems, where these properties often signify deeper relationships or symmetries in the data or equations at hand.