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

Verified step by step guidance

1. 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.
2. Start by finding \( f(-x) \) for the given function \( f(x) = 0.5x^4 - 2x^2 + 6 \). Substitute \( -x \) into the function:
3. \[ f(-x) = 0.5(-x)^4 - 2(-x)^2 + 6 \]
4. 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, \( f(-x) = 0.5x^4 - 2x^2 + 6 \).
5. Compare \( f(-x) \) with \( f(x) \). Since \( f(-x) = f(x) \), the function is even.

Key Concepts

Even and Odd Functions

An even function satisfies the condition 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.

Function Substitution

To test if a function is even or odd, substitute -x into the function in place of x. Simplify the resulting expression and compare it to the original function f(x) and its negative -f(x). This substitution helps verify the symmetry properties of the function.

Polynomial Functions and Symmetry

Polynomial functions with only even powers of x tend to be even functions, while those with only odd powers tend to be odd functions. Mixed powers usually result in neither even nor odd. Recognizing the powers of x in the polynomial helps predict the function's symmetry.