Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=-x^3+2x

Key Concepts

Even Functions

A function is considered 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. A common example of an even function is f(x) = x^2, where substituting -x yields the same result as substituting x.

Odd Functions

A function is classified as 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. An example of an odd function is f(x) = x^3, where substituting -x results in the negative of the function's value at x.

Neither Even Nor Odd Functions

A function is neither even nor odd if it does not satisfy the conditions for either classification. This can occur when the function has terms that do not exhibit symmetry, such as a combination of even and odd powers. For instance, the function f(x) = x^2 + x is neither even nor odd, as it does not meet the criteria for either symmetry.