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

Verified step by step guidance

1. Step 1: Understand the definitions. A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain. A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.
2. Step 2: Substitute \( -x \) into the function \( f(x) = 0.5x^4 - 2x^2 + 6 \) to find \( f(-x) \).
3. Step 3: Calculate \( f(-x) = 0.5(-x)^4 - 2(-x)^2 + 6 \). Simplify each term: \((-x)^4 = x^4\) and \((-x)^2 = x^2\).
4. Step 4: Simplify \( f(-x) \) to get \( f(-x) = 0.5x^4 - 2x^2 + 6 \).
5. Step 5: Compare \( f(-x) \) with \( f(x) \). Since \( f(-x) = f(x) \), the function is even.

Key Concepts

Even and Odd Functions

Even functions are symmetric about the y-axis, meaning that f(x) = f(-x) for all x in the domain. Odd functions have rotational symmetry about the origin, satisfying the condition f(-x) = -f(x). A function can also be neither even nor odd if it does not meet either condition.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers and multiplied by coefficients. The degree of the polynomial, determined by the highest power of x, influences its behavior and symmetry. In this case, the function is a polynomial of degree 4.

Testing for Symmetry

To determine if a function is even or odd, substitute -x into the function and simplify. If the result equals the original function, it is even; if it equals the negative of the original function, it is odd. If neither condition holds, the function is classified as neither even nor odd.