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 1.37.1
In Exercises 37 and 38, write a piecewise formula for the function.
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Verified step by step guidance1
Identify the different intervals on the x-axis where the function changes its behavior. This is typically where the graph has breaks or changes direction.
For each interval, determine the formula that represents the function. This involves analyzing the graph and identifying the type of function (e.g., linear, quadratic) that fits the segment.
Write the piecewise function using the identified formulas for each interval. Use the format: f(x) = {formula1 for interval1, formula2 for interval2, ...}.
Ensure that the conditions for each interval are clearly defined, using inequalities to specify where each formula applies.
Double-check the continuity and endpoints of each interval to ensure the piecewise function accurately represents the graph across all specified intervals.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Piecewise Functions
A piecewise function is defined by different expressions based on the input value's range. Each 'piece' of the function applies to a specific interval of the domain. Understanding how to construct and interpret these functions is crucial for solving problems that involve different behaviors over different intervals.
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05:36
Piecewise Functions
Domain and Range
The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For piecewise functions, it's essential to identify the domain for each piece to ensure the function is correctly defined over its entire range. This helps in writing accurate piecewise formulas.
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5:10Finding the Domain and Range of a Graph
Function Notation
Function notation involves expressing functions in a form like f(x) to denote the output of function f for input x. In piecewise functions, notation is used to specify different expressions for different intervals, often using conditions like 'if' or 'for' to clarify which expression applies to which part of the domain.
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04:22Sigma Notation
Related Practice
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x⁵ - x³ - x
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Textbook Question
Functions and Graphs
Find the natural domain and graph the functions in Exercises 15–20.
g(x) = √−x
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Textbook Question
Shifting Graphs
The accompanying figure shows the graph of y = −x² shifted to four new positions. Write an equation for each new graph.
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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
____
𝔂 = -2 + √1 - x
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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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