Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
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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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