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 \( f(-x) = -f(x) \).
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 \( y(x) = x \cos x \). 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.