Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=-x³+2x

Verified step by step guidance

1. Recall the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \), and it is odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition holds, the function is neither even nor odd.
2. Start by finding \( f(-x) \) for the given function \( f(x) = -x^3 + 2x \). Substitute \( -x \) into the function: \( f(-x) = -(-x)^3 + 2(-x) \).
3. Simplify the expression for \( f(-x) \): \( (-x)^3 = -x^3 \), so \( f(-x) = -(-x^3) + (-2x) = x^3 - 2x \).
4. Compare \( f(-x) = x^3 - 2x \) with \( f(x) = -x^3 + 2x \) and \( -f(x) = x^3 - 2x \). Notice that \( f(-x) = -f(x) \), which matches the condition for an odd function.
5. Conclude that the function \( f(x) = -x^3 + 2x \) is an odd function because it satisfies \( f(-x) = -f(x) \).

Key Concepts

Even and Odd Functions

Even functions satisfy the condition f(-x) = f(x) for all x in their domain, meaning their graphs are symmetric about the y-axis. Odd functions satisfy f(-x) = -f(x), showing symmetry about the origin. Determining whether a function is even, odd, or neither involves testing these conditions.

Function Substitution

To test if a function is even or odd, substitute -x into the function in place of x. Simplify the resulting expression and compare it to the original function f(x) and its negative -f(x). This substitution is essential to verify the symmetry properties of the function.

Polynomial Functions and Their Symmetry

Polynomial functions can be classified as even, odd, or neither based on the powers of x. Terms with even powers contribute to even symmetry, while terms with odd powers contribute to odd symmetry. A polynomial with a mix of even and odd powers is generally neither even nor odd.