Problem 55 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=3 x^{3}-x$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Define Even and Odd Functions

A function is even if \( f(-x) = f(x) \) for every \( x \) in the domain of \( f \). This means the function is symmetric about the y-axis. A function is odd if \( f(-x) = -f(x) \) for every \( x \) in the domain. This means the function is symmetric about the origin.

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

The given function is \( f(x) = 3x^3 - x \). Replace every \( x \) with \( -x \) in the expression to find \( f(-x) \):\[ f(-x) = 3(-x)^3 - (-x) = 3(-x^3) + x \].Simplify the equation: \[ f(-x) = -3x^3 + x \].

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

Firstly, calculate \( -f(x) \):\[-f(x) = -(3x^3 - x) = -3x^3 + x.\]Compare \( f(-x) \) and \( -f(x) \):\( f(-x) = -3x^3 + x \) and \( -f(x) = -3x^3 + x \).Since \( f(-x) = -f(x) \), \( f(x) \) is an odd function.

Key Concepts

Function Symmetry

Function symmetry is all about how a function behaves when it is 'flipped' in different ways. Think of it as a type of mirror test for functions.
A function's symmetry is described mainly in two ways:

• **Symmetry about the y-axis (even functions):** If you can fold a graph along the y-axis and the two sides match perfectly, the graph is symmetric about the y-axis.
• **Symmetry about the origin (odd functions):** Imagine rotating the graph by 180 degrees around the origin. If it looks the same, it is symmetric about the origin.

Understanding function symmetry helps in quickly identifying the nature of the function without having to plot it completely. This makes it invaluable for solving equations and analyzing graphs efficiently.

Odd Functions

Odd functions have a unique symmetry called "origin symmetry," which means they mirror around both the x-axis and y-axis combined. For a function to be odd, the relation needs to satisfy: \( f(-x) = -f(x) \).
Think of odd functions as having a special balancing point at the origin. For instance, in the given problem:- The function is defined as \( f(x) = 3x^3 - x \).- When you replace \( x \) with \( -x \), and simplify, you get \( f(-x) = -3x^3 + x \). - Calculating \( -f(x) \) gives the same result: \( -f(x) = -3x^3 + x \).Since \( f(-x) \) equals \(-f(x)\), this confirms that the function is odd.
Odd functions are useful because they exhibit this symmetry, simplifying calculus and functional analysis tasks.

Even Functions

Even functions display a symmetric pattern around the y-axis. This symmetry can be visualized as both sides of the y-axis looking like mirror images. For a function to qualify as even, it must satisfy the condition: \( f(-x) = f(x) \).
This means no matter where you look at the graph, if you move equally left and right along the horizontal (x-axis), the function values are identical. For example:- If you have a function \( f(x) = x^2 \), then \( f(-x) = (-x)^2 = x^2 \), adhering to \( f(-x) = f(x) \).- Recognizing even functions is beneficial, particularly in areas like geometry and trigonometry where symmetry plays a significant role.They are foundational in many applications, such as simplifying integrals in calculus and foresee functional behavior without exhaustive calculation.