Problem 77 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The function is even.

Step 1: Define Even and Odd Functions

A function is **even** if for every \( x \) in its domain, \( f(x) = f(-x) \). A function is **odd** if for every \( x \) in its domain, \( f(-x) = -f(x) \). If neither condition is satisfied, the function is neither even nor odd.

Step 2: Substitute \(-x\) into the Function

We start by substituting \(-x\) into the function \( f(x) = |x| \cos x \). This gives: \[ f(-x) = |-x| \cos(-x) \] Since the absolute value of a negative number is positive, \(|-x| = |x|\). Also, cosine is an even function, so \(\cos(-x) = \cos x\). Therefore, \[ f(-x) = |x| \cos x \]

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

The expression for \( f(-x) \) matches exactly with \( f(x) \), which is \(|x| \cos x\). Therefore, \( f(-x) = f(x) \). Since this condition satisfies the definition of an even function, \( f(x) \) is classified as even.

Step 4: Conclude the Function's Nature

Because \( f(-x) = f(x) \) is true for \( f(x) = |x| \cos x \), we conclude that the function is even.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics, symbolized as \(|x|\). It describes the distance of a number from zero on the number line, regardless of direction. This means that for any real number \(x\), the absolute value function returns \(x\) if \(x\) is positive or zero and \(-x\) if \(x\) is negative. \[\]

• Mathematically: \(|x| = x\) if \(x \geq 0\); \(|x| = -x\) if \(x < 0\).

One important characteristic of the absolute value function is that it is always non-negative. This means it maps all real numbers to the set of non-negative real numbers. \[\]In the context of function symmetry, when you process \(-x\) through the function \(|x|\), it treats \(-x\) the same way it treats \(x\). So, \(|-x| = |x|\), proving a crucial property that affects whether the whole function can be even. \[\]When combined with other functions like the cosine function, it ensures that outputs essentially flip any input direction back to non-negative. This influences the symmetry properties of composite functions like \(f(x) = |x| \cos x\).

Cosine Function

The cosine function is one of the basic trigonometric functions, often denoted as \(\cos x\). It is periodic, repeating its values every \(2\pi\). A key property of the cosine function is its evenness, meaning \(\cos(-x) = \cos x\). This property directly influences how the function behaves in symmetry tests. \[\]

• Cosine is symmetric across the y-axis (even function).
• Its range is between \(-1\) and \(1\).
• Its graph is a wave-like pattern that repeats every \(2\pi\).

Because the cosine function is even, it helps maintain or cause symmetry when combined with other functions, such as the absolute value function in \(f(x) = |x| \cos x\). Connection with even and odd properties comes into play significantly when determining if a composite function maintains or breaks symmetry. In our example function, the cosine part does not change when you replace \(x\) with \(-x\). This property significantly contributes to determining the whole function's nature as even.

Function Symmetry

Function symmetry is a concept that helps us understand how a function behaves when its inputs are transformed, usually with negative inputs, reflected on the graph. Symmetries can be categorized into different types, such as even, odd, or neither, affecting how functions appear on a graph. \[\]

• An **even function** has symmetry across the y-axis. It satisfies \(f(x) = f(-x)\).
• An **odd function** has symmetry around the origin, meeting the condition \(f(-x) = -f(x)\).
• A function classified as **neither** does not satisfy the criteria for either even or odd.

In the case of the function \(f(x) = |x| \cos x\), it's found to be even as it meets the condition \(f(-x) = f(x)\). This conclusion was reached by substituting \(-x\) into the function and finding that the expression remains the same. The graph of an even function will be a mirror image on either side of the y-axis, which is visually evident when examining both the absolute value function and the cosine function properties. Understanding these symmetries can help in anticipating the graph shape without plotting numerous points.