The graph of the rational function f(x) is shown below. Using the graph, determine which of the following loca behaviors are correct. -5 0. -1- Select all correct answers. Select all that apply: O As x -0o.f(x) 2 O As x - -o.f(x) 2 O As x - -4*.fx) o0 O As x - -1*./x)--00 O As x -1.fo-00 U As x-4./m -0
The graph of the rational function f(x) is shown below. Using the graph, determine which of the following loca behaviors are correct. -5 0. -1- Select all correct answers. Select all that apply: O As x -0o.f(x) 2 O As x - -o.f(x) 2 O As x - -4*.fx) o0 O As x - -1*./x)--00 O As x -1.fo-00 U As x-4./m -0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
The graph of the rational function f(x) is shown below. Using the graph, determine which of the following local and end behaviors are correct.
As x→∞, f(x)→2
As x→−∞, f(x)→2
As x→−4+, f(x)→∞
As x→−1+, f(x)→−∞
As x→−1+, f(x)→∞
As x→−4−, f(x)→−∞
![**Graph Analysis Exercise: Rational Function Behavior**
The graph of a rational function \(f(x)\) is shown below. Using the graph, determine which of the following local behaviors are correct.
![Graph of \(f(x)\)]
Explanation:
The graph presents a rational function with distinct vertical and horizontal asymptotes. By analyzing the graph, we can gain insights into the behavior of the function at various critical points and intervals.
### Analysis Points:
1. **Horizontal Asymptote:** This line suggests that as \( x \) approaches infinity or negative infinity, \( f(x) \) settles towards a constant value.
2. **Vertical Asymptotes:** These lines suggest that the function \( f(x) \) tends to infinity or negative infinity near specific values of \( x \).
### Select all correct answers:
**Which of the following statements about the local behaviors of the function are correct?**
- [ ] As \( x \to \infty, f(x) \to 2 \)
- [ ] As \( x \to -\infty, f(x) \to 2 \)
- [ ] As \( x \to -4^+, f(x) \to \infty \)
- [ ] As \( x \to -1^-, f(x) \to -\infty \)
- [ ] As \( x \to -1^+, f(x) \to \infty \)
- [ ] As \( x \to -4^-, f(x) \to -\infty \)
### Explanation of Graph Features:
- **Horizontal Asymptote at \( y=2 \)**:
- As \( x \to \infty \), the function \( f(x) \) approaches the value 2.
- As \( x \to -\infty \), the function \( f(x) \) also approaches the value 2.
- **Vertical Asymptotes:**
- As \( x \to -4^+ \) (from the right), the function \( f(x) \) approaches \(\infty\).
- As \( x \to -4^- \) (from the left), the function \( f(x) \) approaches \( -\infty \).
- As \( x \to -1^- \) and \( x \to -1^+ \) (from the left and right, respectively), the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a313e2a-25da-4b9a-9850-84d0280d86f1%2F23270166-38b0-43f9-b124-5e002d5ffe68%2Fnho9fnf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graph Analysis Exercise: Rational Function Behavior**
The graph of a rational function \(f(x)\) is shown below. Using the graph, determine which of the following local behaviors are correct.
![Graph of \(f(x)\)]
Explanation:
The graph presents a rational function with distinct vertical and horizontal asymptotes. By analyzing the graph, we can gain insights into the behavior of the function at various critical points and intervals.
### Analysis Points:
1. **Horizontal Asymptote:** This line suggests that as \( x \) approaches infinity or negative infinity, \( f(x) \) settles towards a constant value.
2. **Vertical Asymptotes:** These lines suggest that the function \( f(x) \) tends to infinity or negative infinity near specific values of \( x \).
### Select all correct answers:
**Which of the following statements about the local behaviors of the function are correct?**
- [ ] As \( x \to \infty, f(x) \to 2 \)
- [ ] As \( x \to -\infty, f(x) \to 2 \)
- [ ] As \( x \to -4^+, f(x) \to \infty \)
- [ ] As \( x \to -1^-, f(x) \to -\infty \)
- [ ] As \( x \to -1^+, f(x) \to \infty \)
- [ ] As \( x \to -4^-, f(x) \to -\infty \)
### Explanation of Graph Features:
- **Horizontal Asymptote at \( y=2 \)**:
- As \( x \to \infty \), the function \( f(x) \) approaches the value 2.
- As \( x \to -\infty \), the function \( f(x) \) also approaches the value 2.
- **Vertical Asymptotes:**
- As \( x \to -4^+ \) (from the right), the function \( f(x) \) approaches \(\infty\).
- As \( x \to -4^- \) (from the left), the function \( f(x) \) approaches \( -\infty \).
- As \( x \to -1^- \) and \( x \to -1^+ \) (from the left and right, respectively), the
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