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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![### Question 14: State the Domain of the Following Function
**Graph Description:**
The provided graph illustrates a function with significant features to be examined:
- The x-axis ranges from -6 to 6, while the y-axis ranges from -4 to 4.
- There is a vertical asymptote at \(x = -2\), suggesting that the function is undefined there.
- The graph shows that the function approaches \(y = \infty\) as \(x\) approaches -2 from the left, and approaches \(y = -\infty\) as \(x\) approaches -2 from the right.
- The function is continuous from \(x = -6\) to \(x = -2\), excluding \(x = -2\).
- For \(x > -2\), the function resumes and is continuous, with an open circle (indicating discontinuity) at \((0, 0)\).
- The function is defined at \( (1, 3) \), marked by a solid dot at this coordinate.
#### Domain:
The domain of the function includes all real numbers except \(x = -2\) due to the vertical asymptote, and \(x = 0\) because of the open circle (hole in the graph).
**Mathematically, the domain is:**
\[ (-\infty, -2) \cup (-2, 0) \cup (0, \infty) \]
This covers the intervals where the function is defined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8fa7b6df-3e8d-4d98-b45c-f401a9067270%2F52696077-ab4f-4d82-9ab1-efb5bef67d57%2Fknk3q8b_processed.png&w=3840&q=75)
Transcribed Image Text:### Question 14: State the Domain of the Following Function
**Graph Description:**
The provided graph illustrates a function with significant features to be examined:
- The x-axis ranges from -6 to 6, while the y-axis ranges from -4 to 4.
- There is a vertical asymptote at \(x = -2\), suggesting that the function is undefined there.
- The graph shows that the function approaches \(y = \infty\) as \(x\) approaches -2 from the left, and approaches \(y = -\infty\) as \(x\) approaches -2 from the right.
- The function is continuous from \(x = -6\) to \(x = -2\), excluding \(x = -2\).
- For \(x > -2\), the function resumes and is continuous, with an open circle (indicating discontinuity) at \((0, 0)\).
- The function is defined at \( (1, 3) \), marked by a solid dot at this coordinate.
#### Domain:
The domain of the function includes all real numbers except \(x = -2\) due to the vertical asymptote, and \(x = 0\) because of the open circle (hole in the graph).
**Mathematically, the domain is:**
\[ (-\infty, -2) \cup (-2, 0) \cup (0, \infty) \]
This covers the intervals where the function is defined.
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