Problem 75 - Precalculus Mathematics for Calculus

Question

Determine whether the function is even, odd, or neither. $$f(x)=\sin x \cos x$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Define Properties of Even and Odd Functions

For a function to be even, it must satisfy the condition \( f(-x) = f(x) \) for all \( x \). For a function to be odd, the condition \( f(-x) = -f(x) \) must hold true for all \( x \). If neither condition is met, the function is neither even nor odd.

Step 2: Substitute -x into the Function

Substitute \(-x\) into the function: \( f(-x) = \sin(-x) \cdot \cos(-x) \). Recall that \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \). Therefore, \( f(-x) = -\sin(x) \cdot \cos(x) \).

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

From step 2, we have \( f(-x) = -\sin(x) \cdot \cos(x) \) and the original function \( f(x) = \sin(x) \cdot \cos(x) \).We observe that\( f(-x) = -f(x) \).

Step 4: Conclude the Nature of the Function

Since \( f(-x) = -f(x) \), the function \( f(x) = \sin(x) \cos(x) \) meets the condition for an odd function. Therefore, the function is odd.

Key Concepts

Even and Odd Functions

Understanding whether a function is even, odd, or neither is essential in precalculus. This concept helps in determining the symmetry of the function's graph. Here’s how you identify these functions:

• Even Functions: A function is even if for every x, the equation \( f(-x) = f(x) \) holds true. This means that the graph of an even function is symmetrical about the y-axis.
• Odd Functions: Conversely, a function is odd if for all x, \( f(-x) = -f(x) \) is satisfied. An odd function’s graph has rotational symmetry about the origin.
• Neither: If a function does not satisfy either condition, it is considered neither even nor odd.

The function in the exercise, \( f(x) = \sin(x)\cos(x) \), is found to be odd because replacing x with -x gives \( f(-x) = -f(x) \). This makes it meet the definition of an odd function.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental to mathematics, describing periodic phenomena.

• Sine Function: Denoted as \( \sin(x) \), it is an odd function. This means that \( \sin(-x) = -\sin(x) \).
• Cosine Function: Represented as \( \cos(x) \), it is an even function since \( \cos(-x) = \cos(x) \).
• Tangent Function: Known as \( \tan(x) \), is an odd function, similar to sine.

When analyzing \( f(x) = \sin(x)\cos(x) \), notice that it combines both sine and cosine. The sine part adheres to its odd function property, creating a negative product when x is replaced by -x, which helps determine the odd nature of the entire function.

Function Properties

Function properties aid in understanding more about a function's behavior without graphing it. These properties extend beyond even and odd characteristics:

• Periodicity: Many trigonometric functions, like sine and cosine, repeat their values after a certain interval. This is the function’s period. For instance, \( \sin(x) \) and \( \cos(x) \) have periods of \( 2\pi \).
• Symmetry: Knowing if a function is even or odd can help quickly identify symmetry, which simplifies graphing and understanding the function’s graph.
• Amplitude and Phase Shift: In trigonometric contexts, the amplitude indicates the height of function peaks, while phase shift describes horizontal shifts on a graph. Functions like \( \sin(x) \) often have these characteristics.

These properties describe how a function behaves and interacts, contributing to a clearer understanding of how it operates within its domain. Identifying even and odd properties in functions can simplify complex analysis, making it easier to predict and manipulate function behavior.