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

𝔂 = sec x tan x

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First, recall the definitions of even and odd functions. 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 = sec(x) tan(x) is even, odd, or neither, we need to evaluate y(-x) and compare it to y(x).

Calculate y(-x): Substitute -x into the function to get y(-x) = sec(-x) tan(-x).

Use the trigonometric identities: sec(-x) = sec(x) and tan(-x) = -tan(x). Substitute these into y(-x) to get y(-x) = sec(x) (-tan(x)).

Compare y(-x) = sec(x) (-tan(x)) with y(x) = sec(x) tan(x). Since y(-x) = -y(x), the function y = sec(x) tan(x) is odd.

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Even and Odd Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, a function is odd if it meets the condition f(-x) = -f(x), indicating symmetry about the origin. Understanding these definitions is crucial for determining the nature of the given function.

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Properties of Functions

Trigonometric Functions

The function in question, 𝔶 = sec x tan x, involves trigonometric functions. The secant function, sec x, is defined as 1/cos x, and the tangent function, tan x, is defined as sin x/cos x. Familiarity with the properties and behaviors of these functions is essential for analyzing their symmetry.

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Introduction to Trigonometric Functions

Function Composition and Transformation

To determine if the function is even or odd, one must evaluate the function at -x, which involves substituting -x into the function and simplifying. This process of function composition and transformation is key to analyzing the symmetry properties of the function, allowing for a clear conclusion about its classification.

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