he figure below shows the graph of a rational function f., t has vertical asymptotes x= -1 and x=-5, and horizontal asymptote y=0. the graph does not have an x-intercept, and it passes through the point (-4,-2). The equation for f(x) has one of the five forms shown below. Choose the appropriate form for f(x), and then write the equation. You can assume that f(x) is in simplest form. Of(x) Of(x) = of(x) = Of(x) = Of(x) = a x - b a (x - b) x - c a (x - b)(x - c) a (x - b) (x - c)(x - d) a (x - b)(x - c) (x - d) (x e) = 11 || -O IC 0 0 OC ПОТ OC O OC
he figure below shows the graph of a rational function f., t has vertical asymptotes x= -1 and x=-5, and horizontal asymptote y=0. the graph does not have an x-intercept, and it passes through the point (-4,-2). The equation for f(x) has one of the five forms shown below. Choose the appropriate form for f(x), and then write the equation. You can assume that f(x) is in simplest form. Of(x) Of(x) = of(x) = Of(x) = Of(x) = a x - b a (x - b) x - c a (x - b)(x - c) a (x - b) (x - c)(x - d) a (x - b)(x - c) (x - d) (x e) = 11 || -O IC 0 0 OC ПОТ OC O OC
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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![### Understanding the Graph of a Rational Function
The figure below shows the graph of a rational function \( f \). The graph has the following characteristics:
1. **Vertical Asymptotes**: \( x = -1 \) and \( x = -5 \)
2. **Horizontal Asymptote**: \( y = 0 \)
3. **No \( x \)-intercept**
4. **Passes through the point**: \((-4, -2)\)
You need to find the equation of \( f(x) \) in its simplest form. Choose the appropriate form for \( f(x) \) from one of the five options provided and then write the equation.
### Graph Explanation
The graph consists of two vertical asymptotes at \( x = -1 \) and \( x = -5 \), which means the function approaches infinity at these values of \( x \). There is a horizontal asymptote at \( y = 0 \), indicating that as \( x \) approaches infinity, \( f(x) \) approaches 0. The graph passes through the point \((-4, -2)\), meaning when \( x = -4 \), \( f(x) = -2 \).
### Potential Forms for \( f(x) \)
1. \( \frac{a}{x-b} \)
2. \( \frac{a(x-b)}{x-c} \)
3. \( \frac{a}{(x-b)(x-c)} \)
4. \( \frac{a(x-b)}{(x-c)(x-d)} \)
5. \( \frac{a(x-b)(x-c)}{(x-d)(x-e)} \)
From the given graph characteristics, it is clear that \( f(x) \) has two vertical asymptotes and no \( x \)-intercept, so the best choice must account for these features.
### Equation Determination
Since the given points and asymptotes must be satisfied, you need to determine which function form meets all these criteria. After analyzing options, we find:
\[ f(x) = \frac{a}{(x+1)(x+5)} \]
To satisfy the point \((-4, -2)\):
\[ f(-4) = \frac{a}{(-4+1)(-4+5)} = \frac{a}{(-3)(1)} = \frac{a}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbae73a37-719e-4892-8745-1f816a181ea7%2Fd2dee546-cc88-458d-8c85-0d21581a5f40%2Fgzcm7df_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Graph of a Rational Function
The figure below shows the graph of a rational function \( f \). The graph has the following characteristics:
1. **Vertical Asymptotes**: \( x = -1 \) and \( x = -5 \)
2. **Horizontal Asymptote**: \( y = 0 \)
3. **No \( x \)-intercept**
4. **Passes through the point**: \((-4, -2)\)
You need to find the equation of \( f(x) \) in its simplest form. Choose the appropriate form for \( f(x) \) from one of the five options provided and then write the equation.
### Graph Explanation
The graph consists of two vertical asymptotes at \( x = -1 \) and \( x = -5 \), which means the function approaches infinity at these values of \( x \). There is a horizontal asymptote at \( y = 0 \), indicating that as \( x \) approaches infinity, \( f(x) \) approaches 0. The graph passes through the point \((-4, -2)\), meaning when \( x = -4 \), \( f(x) = -2 \).
### Potential Forms for \( f(x) \)
1. \( \frac{a}{x-b} \)
2. \( \frac{a(x-b)}{x-c} \)
3. \( \frac{a}{(x-b)(x-c)} \)
4. \( \frac{a(x-b)}{(x-c)(x-d)} \)
5. \( \frac{a(x-b)(x-c)}{(x-d)(x-e)} \)
From the given graph characteristics, it is clear that \( f(x) \) has two vertical asymptotes and no \( x \)-intercept, so the best choice must account for these features.
### Equation Determination
Since the given points and asymptotes must be satisfied, you need to determine which function form meets all these criteria. After analyzing options, we find:
\[ f(x) = \frac{a}{(x+1)(x+5)} \]
To satisfy the point \((-4, -2)\):
\[ f(-4) = \frac{a}{(-4+1)(-4+5)} = \frac{a}{(-3)(1)} = \frac{a}{
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