Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^3-x+9

Verified step by step guidance

1. To determine if a function is even, odd, or neither, we need to check the symmetry of the function. For a function \( f(x) \), it is even if \( f(-x) = f(x) \) for all \( x \), and it is odd if \( f(-x) = -f(x) \) for all \( x \).
2. Start by finding \( f(-x) \) for the given function \( f(x) = x^3 - x + 9 \). Substitute \( -x \) into the function: \( f(-x) = (-x)^3 - (-x) + 9 \).
3. Simplify \( f(-x) \): \((-x)^3 = -x^3\) and \(-(-x) = x\), so \( f(-x) = -x^3 + x + 9 \).
4. Compare \( f(-x) = -x^3 + x + 9 \) with \( f(x) = x^3 - x + 9 \). Since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \), the function is neither even nor odd.
5. Conclude that the function \( f(x) = x^3 - x + 9 \) is neither even nor odd based on the symmetry tests.

Key Concepts

Even Functions

A function is classified as 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. A common example of an even function is f(x) = x^2, where substituting -x yields the same output as substituting x.

Odd Functions

A function is considered 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) = x^3, where substituting -x results in the negative of the output for x.

Neither Even Nor Odd Functions

A function is classified as neither even nor odd if it does not satisfy the conditions for either classification. This means that f(-x) does not equal f(x) and also does not equal -f(x). An example would be f(x) = x^3 - x + 9, as it does not exhibit symmetry about the y-axis or the origin.