Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = sec x tan x
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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 1.1.60
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
sin x²
Verified step by step guidance1
Understand the definitions: A function f(x) is even if f(-x) = f(x) for all x in the domain, and odd if f(-x) = -f(x) for all x in the domain.
Consider the function given: f(x) = sin(x²). We need to check the behavior of f(-x) compared to f(x).
Calculate f(-x): Substitute -x into the function to get f(-x) = sin((-x)²). Since (-x)² = x², this simplifies to sin(x²).
Compare f(-x) and f(x): We find that f(-x) = sin(x²) = f(x), which matches the condition for an even function.
Conclude: Since f(-x) = f(x), the function sin(x²) is even. It does not satisfy the condition for an odd function, which would require f(-x) = -f(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even Functions
A function is classified as 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. Common examples include polynomial functions with only even powers, such as f(x) = x².
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Exponential Functions
Odd Functions
A function is considered 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. Typical examples include polynomial functions with only odd powers, such as f(x) = x³.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, have specific properties regarding their symmetry. The sine function is an odd function, meaning sin(-x) = -sin(x). Understanding these properties is crucial when analyzing functions like sin(x²) to determine their evenness or oddness.
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Related Practice
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
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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) = x/(x² − 1)
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Textbook Question
Increasing and Decreasing Functions
Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
y = −x³
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Textbook Question
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (A − B) = sin A cos B − cos A sin B
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