Problem 72 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. The function \(f(x)=|\sin x|\) is even.

Step-by-Step Solution

Answer

True, the function \(f(x) = |\sin x|\) is even as it satisfies \(f(-x) = f(x)\).

Step 1: Understand the Definition of Even Function

A function \(f(x)\) is considered even if it satisfies the condition \(f(-x) = f(x)\) for all \(x\) in the domain of \(f\). This means the function is symmetric with respect to the y-axis.

Step 2: Consider the Function and Its Domain

The function \(f(x) = |\sin x|\) involves the absolute value of the sine function, \(\sin x\), which is defined for all real numbers, \(x \in \mathbb{R}\).

Step 3: Test the Evenness Condition

Calculate \(f(-x) = |\sin(-x)|\). Since \(-\sin x = \sin(-x)\) is a trigonometric identity, it follows that \(f(-x) = |-\sin x| = |\sin x|\).

Step 4: Verify the Evenness Condition

Compare \(f(-x) = |\sin x|\) with \(f(x) = |\sin x|\). Since \(f(-x) = f(x)\) for all \(x\), the function satisfies the even condition.

Key Concepts

Absolute Value

In mathematics, the absolute value of a number refers to its distance from zero on the number line, without considering its direction. For any real number, the absolute value is always non-negative. For example, the absolute value of both -3 and 3 is 3. It is denoted using vertical bars, like \( |x| \). When dealing with functions, applying the absolute value can change how the function behaves. It often reflects any negative inputs to positive outputs. Consider the function \( f(x) = |\sin x| \). The sine function, \( \sin x \), can take both negative and positive values depending on the input \( x \). By applying the absolute value, every output becomes non-negative, effectively removing any negative values the sine function may produce.

• The absolute value ensures that the function output is always zero or greater.
• This property can give the function a symmetry it didn't have before.

The absolute value is crucial in making \( f(x) = |\sin x| \) an even function by ensuring the outputs are the same for both \( x \) and \( -x \). This reflects a mirror-like symmetry relative to the y-axis.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are foundational in mathematics, especially for modeling periodic phenomena. The sine function, represented as \( \sin x \), oscillates between -1 and 1 as \( x \) varies, creating a wave-like pattern.

• The sine function is periodic with a period of \( 2\pi \), meaning it repeats its values in the same pattern every \( 2\pi \) units.
• It is also an odd function, satisfying the property \( \sin(-x) = -\sin(x) \).

When deriving \( f(x) = |\sin x| \), the absolute value alters the sine function's original odd property. Instead of mirroring about the origin as in \( \sin x \), it reflects below the x-axis to form a positive wave that exhibits even function symmetry. This is why it's critical to understand the basic behavior of sine before analyzing \( |\sin x| \). The alteration of the trigonometric sine function through absolute value transforms its graph's symmetry and enables \( \sin x \) to contribute to forming an even function.

Function Symmetry

Symmetry in functions is a key concept that can help us understand their behavior more deeply. A symmetric function is one that mirrors itself across a certain line, typically the y-axis or the origin, depending on whether it is even or odd.An even function, like \( f(x) = |\sin x| \), satisfies the condition \( f(-x) = f(x) \) for all values in its domain. This creates a y-axis symmetry. In simpler terms, the graph of an even function will look the same on both sides of the y-axis.

• This symmetry means that for any point \( (x, y) \) on the function, the point \( (-x, y) \) will also be on the graph.
• Understanding symmetry helps in solving equations and understanding the nature of the function's graph.

The symmetry of \( |\sin x| \) is achieved by the alteration made by the absolute value, as it takes the negative parts of the sine wave and reflects them above the x-axis. This reflection is essential in proving why \( f(x) \) is an even function. It mimics the original periodicity and symmetry of trigonometric functions while adjusting for the absolute value of the sine function, ensuring identical outputs for equivalent positive and negative \( x \) values.