Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = x³ - x + 9

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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 \( f(x) = x^3 - x + 9 \). Substitute \( -x \) into the function: \( f(-x) = (-x)^3 - (-x) + 9 \).

Simplify \( f(-x) \): \( (-x)^3 = -x^3 \), and \( -(-x) = +x \), so \( f(-x) = -x^3 + x + 9 \).

Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): \( f(x) = x^3 - x + 9 \) and \( -f(x) = -x^3 + x - 9 \). Check if \( f(-x) = f(x) \) or \( f(-x) = -f(x) \).

Since \( f(-x) \) is neither equal to \( f(x) \) nor to \( -f(x) \), conclude that the function \( f(x) = x^3 - x + 9 \) is neither even nor odd.

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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. Functions that do not meet either condition are neither even nor odd.

Function Evaluation and Substitution

To determine if a function is even or odd, substitute -x into the function and simplify. Comparing f(-x) with f(x) and -f(x) helps identify the function's symmetry properties. This process is essential for analyzing polynomial and trigonometric functions.

Polynomial Function Properties

Polynomials can be classified by the parity of their terms: even powers contribute to even functions, odd powers to odd functions. A polynomial with mixed powers or constant terms often results in neither even nor odd. Understanding this helps quickly assess the function's symmetry.