Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Understanding the Domain of Inverse Sine Function
**Question:**
The domain of \( \sin^{-1}(x) \) is \([-1,1]\).
- ( ) True
- ( ) False
**Explanation:**
The inverse sine function, often denoted as \( \sin^{-1}(x) \) or arcsin(x), is defined for values of \( x \) that fall within the interval \([-1, 1]\). This means that it only takes inputs \( x \) where \( x \) lies between -1 and 1, inclusive.
### Key Concepts:
1. **Inverse Sine Function (\( \sin^{-1}(x) \))**:
- **Definition**: The inverse sine function reverses the action of the sine function. It answers the question, "What angle has the sine of \( x \)?"
- **Domain**: The set of all possible input values (\( x \)) for which the function is defined is \([-1, 1]\).
- **Range**: The set of all possible output values is \([- \frac{\pi}{2}, \frac{\pi}{2} ]\).
In summary, the correct answer to the question is **True**, because the domain of \( \sin^{-1}(x) \) is indeed \([-1,1]\).
**Graphical Representation**:
If there was a graph or diagram on the screen, it would typically show the function \( \sin^{-1}(x) \) with the x-axis ranging from -1 to 1 and the y-axis ranging from \(-\frac{\pi}{2} \) to \(\frac{\pi}{2} \).
### Additional Features:
- **Sidebar Navigation**: The icon visible on the left suggests there is a sidebar navigation. Clicking this typically reveals additional options or sections within the interface.
- **Status Bar**: The bottom left indicates current weather conditions (89°F, Partly sunny), which is unrelated to the educational content but adds contextual information about the user's environment.
- **Task Bar**: The usual taskbar at the bottom of the screen shows common applications and system status useful for the user interface navigation.
This interactive question format is an excellent way for learners to engage with the material and test their understanding of the domain of trigonometric functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f130582-8262-45ba-b1cd-6680cc0034f3%2F4b8b3cdb-2b7c-4e96-ae9b-4eec1bf2a58f%2Fr938yy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Domain of Inverse Sine Function
**Question:**
The domain of \( \sin^{-1}(x) \) is \([-1,1]\).
- ( ) True
- ( ) False
**Explanation:**
The inverse sine function, often denoted as \( \sin^{-1}(x) \) or arcsin(x), is defined for values of \( x \) that fall within the interval \([-1, 1]\). This means that it only takes inputs \( x \) where \( x \) lies between -1 and 1, inclusive.
### Key Concepts:
1. **Inverse Sine Function (\( \sin^{-1}(x) \))**:
- **Definition**: The inverse sine function reverses the action of the sine function. It answers the question, "What angle has the sine of \( x \)?"
- **Domain**: The set of all possible input values (\( x \)) for which the function is defined is \([-1, 1]\).
- **Range**: The set of all possible output values is \([- \frac{\pi}{2}, \frac{\pi}{2} ]\).
In summary, the correct answer to the question is **True**, because the domain of \( \sin^{-1}(x) \) is indeed \([-1,1]\).
**Graphical Representation**:
If there was a graph or diagram on the screen, it would typically show the function \( \sin^{-1}(x) \) with the x-axis ranging from -1 to 1 and the y-axis ranging from \(-\frac{\pi}{2} \) to \(\frac{\pi}{2} \).
### Additional Features:
- **Sidebar Navigation**: The icon visible on the left suggests there is a sidebar navigation. Clicking this typically reveals additional options or sections within the interface.
- **Status Bar**: The bottom left indicates current weather conditions (89°F, Partly sunny), which is unrelated to the educational content but adds contextual information about the user's environment.
- **Task Bar**: The usual taskbar at the bottom of the screen shows common applications and system status useful for the user interface navigation.
This interactive question format is an excellent way for learners to engage with the material and test their understanding of the domain of trigonometric functions.
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