Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 4 sin ( 1 )
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 1.1.23a
Functions and Graphs
Graph the following equations and explain why they are not graphs of functions of x.
a. |y| = x
Verified step by step guidance1
First, understand the equation |y| = x. This equation involves the absolute value of y, which means y can be either positive or negative for each value of x.
To graph this equation, consider two separate cases: y = x and y = -x. These represent the two possible values of y for each x.
Graph the line y = x. This is a straight line passing through the origin with a slope of 1, meaning it rises one unit for every unit it moves to the right.
Graph the line y = -x. This is another straight line passing through the origin, but with a slope of -1, meaning it falls one unit for every unit it moves to the right.
Notice that for each x-value, there are two corresponding y-values (one positive and one negative). This violates the definition of a function, which requires each x-value to have exactly one y-value. Therefore, the graph is not a function of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of a Function
A function is a relation that assigns exactly one output value for each input value. In mathematical terms, for a relation to be a function, it must pass the vertical line test, meaning that no vertical line intersects the graph of the relation at more than one point. This ensures that each input corresponds to a unique output.
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Definition of the Definite Integral
Vertical Line Test
The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test is crucial for identifying whether a given relation meets the criteria of a function.
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05:13Slopes of Tangent Lines
Absolute Value Functions
An absolute value function, such as |y| = x, describes a relationship where the output is the non-negative value of the input. This specific equation implies that for each positive x, there are two corresponding y values (y and -y), which violates the definition of a function. Thus, it fails the vertical line test, indicating it is not a function of x.
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06:37Average Value of a Function
Related Practice
Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
________
𝔂 = 5 - √ x² - 2x - 3
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Textbook Question
Functions and Graphs
Graph the following equations and explain why they are not graphs of functions of x.
a. |x| + |y| = 1
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Textbook Question
Composition of Functions
Copy and complete the following table.
a. <IMAGE>
Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 2 + 3x² .
x² + 4
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Textbook Question
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
a. <IMAGE>