Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = tan(2x - π)
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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 1.20
In Exercises 19–32, find the (a) domain and (b) range.
____
𝔂 = -2 + √1 - x
Verified step by step guidance1
Step 1: Identify the expression inside the square root, which is '1 - x'. The square root function is defined only for non-negative values, so set up the inequality 1 - x ≥ 0.
Step 2: Solve the inequality 1 - x ≥ 0 to find the domain of the function. Rearrange the inequality to find the values of x that satisfy it.
Step 3: The domain of the function is the set of x-values for which the expression inside the square root is non-negative. Express this domain in interval notation.
Step 4: To find the range, consider the values that the expression √(1 - x) can take. Since the square root function outputs non-negative values, determine the minimum and maximum values of √(1 - x).
Step 5: The range of the function is determined by the expression -2 + √(1 - x). Calculate the minimum and maximum values of this expression based on the domain found in Step 3, and express the range in interval notation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
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 = -2 + √(1 - x), the expression under the square root must be non-negative, which imposes restrictions on x. Thus, determining the domain involves solving the inequality 1 - x ≥ 0.
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Range of a Function
The range of a function is the set of all possible output values (y-values) that the function can produce. For the function y = -2 + √(1 - x), the square root function outputs non-negative values, which means the minimum value of y occurs when x is at its maximum in the domain. Analyzing the function helps identify the range based on the values y can take.
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Square Root Function
The square root function, denoted as √x, is defined for non-negative values of x and produces non-negative outputs. In the context of the function y = -2 + √(1 - x), the square root affects both the domain and range, as it restricts x to values where 1 - x is non-negative, and it shifts the output down by 2, impacting the overall range.
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Related Practice
Textbook Question
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Find the natural domain and graph the functions in Exercises 15–20.
g(x) = √−x
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Textbook Question
In Exercises 37 and 38, write a piecewise formula for the function.
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Textbook Question
[Technology Exercise]
a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which
3/(x − 1) < 2/(x + 1)
b. Confirm your findings in part (a) algebraically.
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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = |x| - 2
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Textbook Question
Evaluate cos (11π/12) as cos (π/4 + 2π/3).
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