In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x⁵ - x³ - x

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First, recall the definitions: A function f(x) is even if f(-x) = f(x) for all x in the domain, and it is odd if f(-x) = -f(x) for all x in the domain.

To determine if the function y = x⁵ - x³ - x is even, odd, or neither, start by substituting -x into the function: y(-x) = (-x)⁵ - (-x)³ - (-x).

Simplify the expression y(-x): (-x)⁵ = -x⁵, (-x)³ = -x³, and (-x) = -x. Therefore, y(-x) = -x⁵ + x³ + x.

Compare y(-x) = -x⁵ + x³ + x with the original function y = x⁵ - x³ - x. Notice that y(-x) is not equal to y(x) and y(-x) is not equal to -y(x).

Since y(-x) ≠ y(x) and y(-x) ≠ -y(x), the function y = x⁵ - x³ - x is neither even nor odd.

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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. For example, the function f(x) = x² is even because f(-x) = (-x)² = x².

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) = x³, as f(-x) = (-x)³ = -x³.

Neither Even Nor Odd Functions

A function is neither even nor odd if it does not satisfy the conditions for either classification. This means that the function's graph lacks symmetry with respect to both the y-axis and the origin. For instance, the function f(x) = x + 1 is neither even nor odd, as it does not fulfill the criteria for either symmetry.

In Exercises 9–16, determine whether the function is even, odd, or neither.𝔂 = x⁵ - x³ - x