In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x cos x

Verified step by step guidance

1. To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties. A function f(x) is even if f(-x) = f(x) for all x in the domain, and it is odd if f(-x) = -f(x) for all x in the domain.
2. Start by substituting -x into the function y = x cos(x). This gives us y(-x) = (-x) cos(-x).
3. Recall that the cosine function is even, meaning cos(-x) = cos(x). Therefore, y(-x) = (-x) cos(x).
4. Now, compare y(-x) = (-x) cos(x) with the original function y = x cos(x). Notice that y(-x) = -y(x), which satisfies the condition for the function to be odd.
5. Since y(-x) = -y(x), the function y = x cos(x) is classified as an odd function.

Key Concepts

Even Functions

A function is considered even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. For example, the function f(x) = x² is even because f(-x) = (-x)² = x².

Odd Functions

A function is classified as odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, as f(-x) = (-x)³ = -x³.

Function Analysis

Function analysis involves evaluating the properties of a function, such as its symmetry, continuity, and limits. In the context of determining if a function is even, odd, or neither, one typically substitutes -x into the function and compares the result to f(x) and -f(x). This analysis is crucial for understanding the behavior of functions in calculus.