Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = 0.5x⁴ - 2x² + 6

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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.

Start by finding \( f(-x) \) for the given function \( f(x) = 0.5x^{4} - 2x^{2} + 6 \). Substitute \( -x \) into the function:

\[ f(-x) = 0.5(-x)^{4} - 2(-x)^{2} + 6 \]

Simplify each term using the properties of exponents: \( (-x)^{4} = x^{4} \) because an even power makes the negative sign disappear, and \( (-x)^{2} = x^{2} \) for the same reason. So, \( f(-x) = 0.5x^{4} - 2x^{2} + 6 \).

Compare \( f(-x) \) with \( f(x) \). Since \( f(-x) = f(x) \), the function is even.

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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.

Polynomial Function Symmetry

Polynomial functions exhibit symmetry based on the powers of x. Terms with even powers (like x², x⁴) are even functions, while terms with odd powers (like x³, x) are odd functions. The overall function's parity depends on the combination of these terms.

Substitution Method for Testing Parity

To determine if a function is even or odd, substitute -x into the function and simplify. Compare f(-x) to f(x) and -f(x). This method provides a straightforward algebraic test to classify the function's symmetry.