Textbook Question
The Greatest and Least Integer Functions
Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.12
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = sec x tan x
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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6:04Introduction 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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5:25Intro to Transformations
Related Practice
Textbook Question
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
Textbook Question
Functions and Graphs
Find the natural domain and graph the functions in Exercises 15–20.
f(x) = 1 − 2x − x²
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Textbook Question
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
g(x) = x⁴ + 3x² − 1
Textbook Question
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
sin x²
Textbook Question
Graph the function y = √|x|.
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