Problem 76 - Algebra and Trigonometry

Question

Determine whether the function is even, odd, or neither. $$ f(x)=x \sin ^{3} x $$

Step-by-Step Solution

Answer

The function is even.

Step 1: Definition Check

A function is even if for every x in the domain, \( f(x) = f(-x) \). A function is odd if for every x in the domain, \( f(-x) = -f(x) \). Otherwise, it is neither.

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

Substitute \(-x\) into the given function: \( f(-x) = (-x) \sin^3(-x) \). Simplify this expression by applying the property of sine that \( \sin(-x) = -\sin(x) \). Thus, \( \sin^3(-x) = (-\sin(x))^3 = -\sin^3(x) \). This leads to \( f(-x) = -x(-\sin^3(x)) = x \sin^3(x) \).

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

Compare \( f(x) \) with \( f(-x) \): we have \( f(x) = x \sin^3(x) \) and \( f(-x) = x \sin^3(x) \). This shows that \( f(x) = f(-x) \).

Step 4: Determine if the Function is Even, Odd, or Neither

Since \( f(x) = f(-x) \), the function is even according to the definition in Step 1.

Key Concepts

Properties of Sine Function

The sine function, denoted as \( \sin(x) \), is pivotal in trigonometry and appears in numerous mathematical and real-world applications. Let's dive into some of its crucial properties to understand its behavior better.

• Periodicity: The sine function is periodic with a period of \(2\pi\). This means that \(\sin(x) = \sin(x + 2\pi)\) for any value of \(x\). This property helps simplify expressions involving sine over extended intervals.
• Symmetry: Sine exhibits a type of symmetry known as odd symmetry, stating that \(\sin(-x) = -\sin(x)\). This characteristic is essential for exploring function symmetries, as seen in the analysis of functions like \(x \sin^3(x)\).
• Range: The range of \(\sin(x)\) is between \(-1\) and \(1\), inclusive. This constraint is crucial when considering the possible values of functions involving sine.
• Zeros: The sine function equals zero at integer multiples of \(\pi\), i.e., \(\sin(n\pi) = 0\), where \(n\) is an integer. Knowing where functions reach zero can aid in plotting and analyzing them.

Mastering these properties allows for straightforward manipulation and understanding of more complex expressions like \(x \sin^3(x)\), where these characteristics directly influence the function's nature.

Function Symmetry

Understanding symmetry in functions is a key part of analyzing their behavior. Symmetry can simplify calculations and provide insights into the function's graph.

Let's break down the two main types of symmetry that functions can exhibit:

• Even Functions: A function is considered even if for every \(x\) in its domain, \(f(x) = f(-x)\). Graphically, this symmetry appears as a mirror image about the y-axis. A prime example is \(f(x) = x^2\).
• Odd Functions: A function is odd if for every \(x\) in its domain, \(f(-x) = -f(x)\). This form of symmetry results in rotational symmetry of 180 degrees about the origin on a graph, like \(f(x) = x^3\).

In the given problem of \(x \sin^3(x)\), substituting \(-x\) results in the same expression, indicating that \(f(x) = f(-x)\). Thus, the function is even, a vital conclusion derived from symmetry properties of sine and the structure of the function itself.

Domain of a Function

The domain of a function is a set of all possible input values (x-values) that produce real output values. For each function, understanding its domain is crucial since it impacts where the function is defined and potentially affects its behavior.

For the sine function, \(\sin(x)\), the domain is all real numbers, \(x \in \mathbb{R}\), because it can take any real input and provide a real output. This broad domain is significant when sine is a component of larger equations.

When considering a composite function like \(x \sin^3(x)\):

• Variable \(x\): The term \(x\) also doesn't impose any restrictions, allowing it to take any real number.
• Overall Domain: Since neither the sine function nor \(x\) restrict the inputs, the domain remains all real numbers \( \mathbb{R} \). There are no denominators or roots that could limit possible inputs.

Understanding the domain confirms where you can safely evaluate the function without encountering undefined behavior, thus ensuring accurate insights into properties and symmetries.