Problem 65 - Algebra and Trigonometry

Question

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^{3} - {x}$$

Step-by-Step Solution

Answer

The function is odd, with symmetry about the origin.

Step 1: Understand the definitions

To determine if the function \( f(x) = x^3 - x \) is even, odd, or neither, we need to use the definitions: A function is even if \( f(-x) = f(x) \) for all \( x \), and it is odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition holds, then the function is neither even nor odd.

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

Substitute \( -x \) into the function: \( f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) \).

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

We have \( f(-x) = -(x^3 - x) \). Since \( f(-x) = -f(x) \), the function satisfies the condition for being an odd function. Therefore, \( f(x) = x^3 - x \) is an odd function.

Step 4: Use symmetry to sketch the graph

Since the function is odd, its graph is symmetric about the origin. This means if you have a point \((a, b)\) on the graph, then \((-a, -b)\) will also be a point on the graph.

Key Concepts

Function Symmetry

In mathematics, symmetry of functions is an important concept that allows us to understand their behavior more deeply. Symmetry can simplify the analysis and graphing of functions. Generally, there are two types of symmetry we consider: **symmetry about the y-axis** and **symmetry about the origin**. Each type corresponds to different classifications of functions:

• A function that is symmetric with respect to the y-axis is known as an even function. This means the function produces the same outputs for both positive and negative inputs of the same value.
• A function that is symmetric with respect to the origin is known as an odd function. Here, flipping the graph 180 degrees around the origin results in the same graph.

Knowing these symmetries helps us when sketching graphs, as the need to calculate outputs for each point is reduced.

Even and Odd Functions

Even and odd functions are important classifications in function symmetry. Let's explore what these terms mean and how to determine if a function belongs to either category:For a function to be even, it must satisfy the condition:\[ f(-x) = f(x) \]for all values of \(x\). A common example of an even function is \(f(x) = x^2\), as both \(f(x)\) and \(f(-x)\) result in \(x^2\).An odd function must satisfy that:\[ f(-x) = -f(x) \]for all values of \(x\). For instance, \(f(x) = x^3\) is an odd function since substituting \(-x\) results in:\[ f(-x) = (-x)^3 = -x^3 = -f(x). \]If neither condition for even nor odd functions is met, the function is considered neither even nor odd. Determining this is crucial in predicting how a function's graph will behave.

Graph Sketching

Graph sketching can be greatly simplified by understanding the properties of even and odd functions. Identifying these properties enables a quicker and more accurate graph, as certain symmetries provide vital clues about the general shape: - **For even functions**, your sketch should mirror itself on the y-axis. Drawing one side automatically grants knowledge about the other half. This reflection characteristic makes even functions particularly straightforward to graph. - **For odd functions**, the plot should appear symmetric about the origin. If you have a drawing of one part of the graph around the origin, you can easily rotate it 180 degrees to complete the picture. This symmetry aspect is particularly useful in visualizing complex function graphs quickly. In addition to symmetry, identifying critical points, intercepts, and behavior at infinity can assist in sketching functions more accurately. A clear approach and methodical analysis can significantly enhance the overall sketching process.