Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=0.5x^4-2x^2+6

Key Concepts

Even Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. Common examples include polynomial functions with only even powers of x, such as f(x) = x^2.

Odd Functions

A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. Typical examples include polynomial functions with only odd powers of x, like f(x) = x^3.

Neither Even Nor Odd Functions

A function is classified as neither even nor odd if it does not satisfy the conditions for either classification. This can occur when a function contains both even and odd powers of x or has constant terms that disrupt symmetry. An example is f(x) = x^3 + 2, which does not exhibit symmetry about the y-axis or the origin.