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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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 1.1.54
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)
Verified step by step guidance1
To determine if a function is even, odd, or neither, we need to evaluate the function at -x and compare it to the original function g(x).
Calculate g(-x) by substituting -x into the function: g(-x) = (-x)/((-x)² - 1).
Simplify g(-x): Since (-x)² = x², we have g(-x) = -x/(x² - 1).
Compare g(-x) with g(x): g(x) = x/(x² - 1) and g(-x) = -x/(x² - 1). Notice that g(-x) = -g(x).
Since g(-x) = -g(x), the function g(x) is odd. A function is odd if g(-x) = -g(x) for all x in the domain of the function.
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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 an even function is symmetric with respect to the y-axis. A common example of an even function is f(x) = x², where substituting -x yields the same output as substituting x.
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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 an odd function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, where substituting -x results in the negative of the output for x.
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Function Analysis
Analyzing a function involves evaluating its behavior under transformations, such as substituting -x for x. This process helps determine whether the function is even, odd, or neither. For the function g(x) = x/(x² - 1), one must compute g(-x) and compare it to g(x) and -g(x) to classify its symmetry.
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Related Practice
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.
sin x²
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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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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 = x³ Left 1, down 1
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