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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