Determine whether each function is even, odd, or neither. 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 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 similarly \( (-x)^{2} = x^{2} \). 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), showing symmetry about the origin. Determining whether a function is even, odd, or neither involves testing these conditions.

Polynomial Functions and Their Symmetry

Polynomial functions are sums of terms with variables raised to whole number powers. Even-powered terms (like x^2, x^4) are even functions, while odd-powered terms (like x^3, x) are odd functions. Understanding the parity of each term helps analyze the overall function's symmetry.

Substitution Method for Testing Function Parity

To test 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 way to verify the function's symmetry properties and classify it accordingly.

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