Textbook Question
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31β36.
sin (A β B) = sin A cos B β cos A sin B
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 1.14
In Exercises 9β16, determine whether the function is even, odd, or neither.
π = x - sin x
Verified step by step guidance1
Step 1: Understand 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.
Step 2: Substitute -x into the function y = x - sin(x) to find y(-x). This gives y(-x) = -x - sin(-x).
Step 3: Use the property of the sine function that sin(-x) = -sin(x). Substitute this into y(-x) to simplify: y(-x) = -x + sin(x).
Step 4: Compare y(-x) = -x + sin(x) with the original function y = x - sin(x). Check if y(-x) equals y(x) or -y(x).
Step 5: Determine if the function is even, odd, or neither based on the comparison. If y(-x) = y(x), the function is even. If y(-x) = -y(x), the function is odd. If neither condition is met, the function is neither even nor odd.
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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 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. An example of an even function is f(x) = xΒ², where substituting -x yields the same result as substituting x.
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Exponential Functions
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. A classic example of an odd function is f(x) = xΒ³, where substituting -x results in the negative of the function's value at x.
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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 f(-x) does not equal f(x) or -f(x) for all x. An example is f(x) = x - sin(x), as it does not exhibit symmetry about the y-axis or the origin.
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Related Practice
Textbook Question
Even and Odd Functions
In Exercises 47β62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = xΒ² + x
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Textbook Question
In Exercises 19β32, find the (a) domain and (b) range.
_____
π = -1 + β 2 - x
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Textbook Question
Use graphing software to graph the functions specified in Exercises 31β36.
Select a viewing window that reveals the key features of the function.
Graph two periods of the function f (x) = 3 cot (x/2) + 1.
Textbook Question
Even and Odd Functions
In Exercises 47β62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = xβ»β΅
Textbook Question
In Exercises 5β8, determine whether the graph of the function is symmetric about the π-axis, the origin, or neither.
π = xΒ²/β΅
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