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

𝔂 = 1 - cos x

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To determine if a function is even, odd, or neither, we need to check the symmetry properties of the function. 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.

Start by substituting -x into the function y = 1 - cos(x). This gives us y(-x) = 1 - cos(-x).

Recall that the cosine function is an even function, meaning cos(-x) = cos(x). Therefore, y(-x) = 1 - cos(x).

Compare y(-x) = 1 - cos(x) with the original function y = 1 - cos(x). Since y(-x) = y(x), the function is even.

Conclude that the function y = 1 - cos(x) is even because it satisfies the condition for even functions, y(-x) = y(x).

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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) = x^2 is even because f(-x) = (-x)^2 = x^2.

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, as f(-x) = (-x)^3 = -x^3.

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 does not exhibit symmetry about the y-axis or 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 = 1 - cos x