Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
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𝔂 = -1 + ∛ 2 - 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 1.1.48
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x⁻⁵
Verified step by step guidance1
To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties. 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.
Let's start by finding f(-x) for the given function f(x) = x⁻⁵. Substitute -x into the function: f(-x) = (-x)⁻⁵.
Simplify the expression f(-x) = (-x)⁻⁵. Since the exponent is an odd number, we have f(-x) = -x⁻⁵.
Now, compare f(-x) with f(x). We have f(-x) = -x⁻⁵ and f(x) = x⁻⁵. Notice that f(-x) = -f(x), which satisfies the condition for the function to be odd.
Therefore, the function f(x) = x⁻⁵ is an odd function because f(-x) = -f(x) for all x in its domain.
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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. For example, the function f(x) = x² is even because f(-x) = (-x)² = 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. An example of an odd function is f(x) = x³, as f(-x) = (-x)³ = -x³.
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Neither Even Nor Odd Functions
A function is classified as 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 in its domain. For instance, the function f(x) = x + 1 is neither even nor odd, as f(-x) = -x + 1 does not equal f(x) or -f(x).
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Related Practice
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x - sin x
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Textbook Question
Use graphing software to graph the functions specified in Exercises 31–36.
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
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = x²/⁵
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Textbook Question
Shifting Graphs
Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.
y = −√x Right 3
Textbook Question
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph the upper branch of the hyperbola y² − 16x² = 1.