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


𝔂 = sec x tan x

Verified step by step guidance

1. 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.
2. 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).
3. Calculate y(-x): Substitute -x into the function to get y(-x) = sec(-x) tan(-x).
4. 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)).
5. 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.

Key Concepts

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.

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.

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.