Textbook Question
In Exercises 19β32, find the (a) domain and (b) range.
π = tan(2x - Ο)
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.19
In Exercises 19β32, find the (a) domain and (b) range.
π = |x| - 2
Verified step by step guidance1
Step 1: Identify the function type. The given function is π¦ = |π₯| - 2, which is an absolute value function. Absolute value functions are defined for all real numbers.
Step 2: Determine the domain of the function. Since the absolute value function is defined for all real numbers, the domain of π¦ = |π₯| - 2 is all real numbers, which can be expressed as (-β, β).
Step 3: Analyze the transformation of the function. The function π¦ = |π₯| - 2 is a vertical shift of the basic absolute value function π¦ = |π₯|. The graph of π¦ = |π₯| is shifted 2 units downward.
Step 4: Determine the range of the function. The basic absolute value function π¦ = |π₯| has a range of [0, β). After shifting the graph 2 units downward, the range of π¦ = |π₯| - 2 becomes [-2, β).
Step 5: Summarize the domain and range. The domain of the function π¦ = |π₯| - 2 is all real numbers (-β, β), and the range is [-2, β).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function y = |x| - 2, the absolute value function |x| is defined for all real numbers, meaning the domain is all real numbers, or (-β, β). Understanding the domain is crucial for determining the valid inputs for the function.
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Range
The range of a function is the set of all possible output values (y-values) that the function can produce. In the case of y = |x| - 2, the minimum value occurs when |x| is zero, resulting in y = -2. As x increases or decreases, y increases without bound. Therefore, the range is [-2, β), indicating that y can take any value greater than or equal to -2.
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Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This means |x| is always zero or positive. In the function y = |x| - 2, the absolute value affects the shape of the graph, creating a V-like structure that opens upwards, shifted down by 2 units. Understanding this function is essential for analyzing the overall behavior of the given equation.
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Related Practice
Textbook Question
In Exercises 19β32, find the (a) domain and (b) range.
____
π = -2 + β1 - x
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Textbook Question
[Technology Exercise]
a. Graph the functions f(x) = 3/(x β 1) and g(x) = 2/(x + 1) together to identify the values of x for which
3/(x β 1) < 2/(x + 1)
b. Confirm your findings in part (a) algebraically.
Textbook Question
Even and Odd Functions
In Exercises 47β62, say whether the function is even, odd, or neither. Give reasons for your answer.
h(t) = |tΒ³|
Textbook Question
Evaluate cos (11Ο/12) as cos (Ο/4 + 2Ο/3).
Textbook Question
Increasing and Decreasing Functions
Graph the functions in Exercises 37β46. 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 = 1/|x|
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