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

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Recall the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \), and odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition holds, the function is neither even nor odd.

Start by finding \( f(-x) \) for the given function \( f(x) = -x^{3} + 2x \). Substitute \( -x \) into the function: \( f(-x) = -(-x)^{3} + 2(-x) \).

Simplify the expression for \( f(-x) \): Remember that \( (-x)^{3} = -x^{3} \), so \( f(-x) = -(-x^{3}) + (-2x) = x^{3} - 2x \).

Compare \( f(-x) = x^{3} - 2x \) with \( f(x) = -x^{3} + 2x \) and also with \( -f(x) = x^{3} - 2x \). Notice that \( f(-x) = -f(x) \), which matches the condition for an odd function.

Conclude that since \( f(-x) = -f(x) \), the function \( f(x) = -x^{3} + 2x \) is an odd function.

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Even and Odd Functions

An even function satisfies f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), indicating symmetry about the origin. Determining whether a function is even, odd, or neither involves testing these conditions.

Function Substitution and Simplification

To test if a function is even or odd, substitute -x into the function and simplify the expression. Comparing the result with the original function f(x) and its negative -f(x) helps identify the function's symmetry properties. Accurate algebraic manipulation is essential in this step.

Polynomial Function Properties

Polynomial functions have terms with powers of x, where even powers contribute to even symmetry and odd powers contribute to odd symmetry. For example, x³ is an odd function term, while x² is even. Understanding these properties aids in quickly assessing the overall function's parity.