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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![### Problem Statement
**5) Find the vertical asymptote(s) of the graph of the given function.**
\[ f(x) = \frac{x^2 - 100}{(x - 9)(x + 3)} \]
### Explanation
To find the vertical asymptotes of the function \(f(x)\), we need to determine the values of \(x\) for which the function is undefined due to division by zero. These values typically occur where the denominator of the rational function equals zero.
#### Steps to Find Vertical Asymptotes:
1. **Set the denominator equal to zero:**
\[ (x - 9)(x + 3) = 0 \]
2. **Solve for \(x\):**
- \(x - 9 = 0\)
- \(x + 3 = 0\)
3. The solutions to these equations will give us the \(x\)-values where the vertical asymptotes occur:
- Solving \(x - 9 = 0\) gives \(x = 9\).
- Solving \(x + 3 = 0\) gives \(x = -3\).
Therefore, the vertical asymptotes are at:
\[ x = 9 \]
\[ x = -3 \]
### Conclusion
The function \( f(x) \) has vertical asymptotes at \( x = 9 \) and \( x = -3 \).
### Diagram Explanation
In a graph of the function \( f(x) \), vertical asymptotes would appear as vertical lines at \( x = 9 \) and \( x = -3 \). These lines represent values of \( x \) at which the function approaches infinity or negative infinity, indicating points where the function is undefined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a0a3e2b-c166-4815-91b8-41dfa54aca6a%2F3bd14fd7-9d93-4831-b9f5-899cf51773f8%2Fzu5946j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**5) Find the vertical asymptote(s) of the graph of the given function.**
\[ f(x) = \frac{x^2 - 100}{(x - 9)(x + 3)} \]
### Explanation
To find the vertical asymptotes of the function \(f(x)\), we need to determine the values of \(x\) for which the function is undefined due to division by zero. These values typically occur where the denominator of the rational function equals zero.
#### Steps to Find Vertical Asymptotes:
1. **Set the denominator equal to zero:**
\[ (x - 9)(x + 3) = 0 \]
2. **Solve for \(x\):**
- \(x - 9 = 0\)
- \(x + 3 = 0\)
3. The solutions to these equations will give us the \(x\)-values where the vertical asymptotes occur:
- Solving \(x - 9 = 0\) gives \(x = 9\).
- Solving \(x + 3 = 0\) gives \(x = -3\).
Therefore, the vertical asymptotes are at:
\[ x = 9 \]
\[ x = -3 \]
### Conclusion
The function \( f(x) \) has vertical asymptotes at \( x = 9 \) and \( x = -3 \).
### Diagram Explanation
In a graph of the function \( f(x) \), vertical asymptotes would appear as vertical lines at \( x = 9 \) and \( x = -3 \). These lines represent values of \( x \) at which the function approaches infinity or negative infinity, indicating points where the function is undefined.
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