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


𝔂 = x - sin x

Verified step by step guidance

1. Step 1: Understand the definitions of even and odd functions. 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. Step 2: Substitute -x into the function y = x - sin(x) to find y(-x). This gives y(-x) = -x - sin(-x).
3. Step 3: Use the property of the sine function that sin(-x) = -sin(x). Substitute this into y(-x) to simplify: y(-x) = -x + sin(x).
4. Step 4: Compare y(-x) = -x + sin(x) with the original function y = x - sin(x). Check if y(-x) equals y(x) or -y(x).
5. Step 5: Determine if the function is even, odd, or neither based on the comparison. If y(-x) = y(x), the function is even. If y(-x) = -y(x), the function is odd. If neither condition is met, the function is neither even nor odd.

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. An example of an even function is f(x) = x², where substituting -x yields the same result as substituting x.

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. A classic example of an odd function is f(x) = x³, where substituting -x results in the negative of the function's value at x.

Neither Even Nor Odd Functions

A function is neither even nor odd if it does not satisfy the conditions for either classification. This means that f(-x) does not equal f(x) or -f(x) for all x. An example is f(x) = x - sin(x), as it does not exhibit symmetry about the y-axis or the origin.