The image displays a mathematical problem involving the analysis of a rational function: \[ f(x) = \frac{5x + 15}{x + 3} \] The task involves finding the following characteristics of the given function: 1. **Horizontal Asymptote**: - A horizontal asymptote is a horizontal line that the graph of the function approaches but never touches as x approaches infinity or negative infinity. - Placeholder for the equation of the horizontal asymptote: \( y = [\text{Enter value or 'No horizontal asymptote'}] \) 2. **Vertical Asymptote**: - A vertical asymptote is a vertical line that the graph approaches but never touches as the function heads towards infinity at specific x-values. - Placeholder for the equation of the vertical asymptote: \( x = [\text{Enter value or 'No vertical asymptote'}] \) 3. **x-Intercept**: - The x-intercept is the point where the graph crosses the x-axis. This occurs where \( f(x) = 0 \). - Placeholder for the coordinates of the x-intercept: \( ([\text{Enter value}], 0) \) or "No x-intercept". 4. **y-Intercept**: - The y-intercept is the point where the graph crosses the y-axis. This occurs by evaluating \( f(0) \). - Placeholder for the coordinates of the y-intercept: \( (0, [\text{Enter value}]) \) or "No y-intercept". 5. **Hole**: - A hole in the graph occurs when there's a removable discontinuity, meaning a common factor in the numerator and denominator cancels out. - Placeholder for the coordinates of the hole: \( ([\text{Enter x-coordinate}], [\text{Enter y-coordinate}]) \) or "No hole". Each characteristic requires evaluation based on the properties of the rational function given.
The image displays a mathematical problem involving the analysis of a rational function: \[ f(x) = \frac{5x + 15}{x + 3} \] The task involves finding the following characteristics of the given function: 1. **Horizontal Asymptote**: - A horizontal asymptote is a horizontal line that the graph of the function approaches but never touches as x approaches infinity or negative infinity. - Placeholder for the equation of the horizontal asymptote: \( y = [\text{Enter value or 'No horizontal asymptote'}] \) 2. **Vertical Asymptote**: - A vertical asymptote is a vertical line that the graph approaches but never touches as the function heads towards infinity at specific x-values. - Placeholder for the equation of the vertical asymptote: \( x = [\text{Enter value or 'No vertical asymptote'}] \) 3. **x-Intercept**: - The x-intercept is the point where the graph crosses the x-axis. This occurs where \( f(x) = 0 \). - Placeholder for the coordinates of the x-intercept: \( ([\text{Enter value}], 0) \) or "No x-intercept". 4. **y-Intercept**: - The y-intercept is the point where the graph crosses the y-axis. This occurs by evaluating \( f(0) \). - Placeholder for the coordinates of the y-intercept: \( (0, [\text{Enter value}]) \) or "No y-intercept". 5. **Hole**: - A hole in the graph occurs when there's a removable discontinuity, meaning a common factor in the numerator and denominator cancels out. - Placeholder for the coordinates of the hole: \( ([\text{Enter x-coordinate}], [\text{Enter y-coordinate}]) \) or "No hole". Each characteristic requires evaluation based on the properties of the rational function given.
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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can you write out how to solve all of these please?
![The image displays a mathematical problem involving the analysis of a rational function:
\[ f(x) = \frac{5x + 15}{x + 3} \]
The task involves finding the following characteristics of the given function:
1. **Horizontal Asymptote**:
- A horizontal asymptote is a horizontal line that the graph of the function approaches but never touches as x approaches infinity or negative infinity.
- Placeholder for the equation of the horizontal asymptote: \( y = [\text{Enter value or 'No horizontal asymptote'}] \)
2. **Vertical Asymptote**:
- A vertical asymptote is a vertical line that the graph approaches but never touches as the function heads towards infinity at specific x-values.
- Placeholder for the equation of the vertical asymptote: \( x = [\text{Enter value or 'No vertical asymptote'}] \)
3. **x-Intercept**:
- The x-intercept is the point where the graph crosses the x-axis. This occurs where \( f(x) = 0 \).
- Placeholder for the coordinates of the x-intercept: \( ([\text{Enter value}], 0) \) or "No x-intercept".
4. **y-Intercept**:
- The y-intercept is the point where the graph crosses the y-axis. This occurs by evaluating \( f(0) \).
- Placeholder for the coordinates of the y-intercept: \( (0, [\text{Enter value}]) \) or "No y-intercept".
5. **Hole**:
- A hole in the graph occurs when there's a removable discontinuity, meaning a common factor in the numerator and denominator cancels out.
- Placeholder for the coordinates of the hole: \( ([\text{Enter x-coordinate}], [\text{Enter y-coordinate}]) \) or "No hole".
Each characteristic requires evaluation based on the properties of the rational function given.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5467b03-6856-4d2a-8749-1a2422daa076%2F360f2f6a-900b-42d8-b386-f2a4016b5ed0%2Fe8tqr0y.jpeg&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical problem involving the analysis of a rational function:
\[ f(x) = \frac{5x + 15}{x + 3} \]
The task involves finding the following characteristics of the given function:
1. **Horizontal Asymptote**:
- A horizontal asymptote is a horizontal line that the graph of the function approaches but never touches as x approaches infinity or negative infinity.
- Placeholder for the equation of the horizontal asymptote: \( y = [\text{Enter value or 'No horizontal asymptote'}] \)
2. **Vertical Asymptote**:
- A vertical asymptote is a vertical line that the graph approaches but never touches as the function heads towards infinity at specific x-values.
- Placeholder for the equation of the vertical asymptote: \( x = [\text{Enter value or 'No vertical asymptote'}] \)
3. **x-Intercept**:
- The x-intercept is the point where the graph crosses the x-axis. This occurs where \( f(x) = 0 \).
- Placeholder for the coordinates of the x-intercept: \( ([\text{Enter value}], 0) \) or "No x-intercept".
4. **y-Intercept**:
- The y-intercept is the point where the graph crosses the y-axis. This occurs by evaluating \( f(0) \).
- Placeholder for the coordinates of the y-intercept: \( (0, [\text{Enter value}]) \) or "No y-intercept".
5. **Hole**:
- A hole in the graph occurs when there's a removable discontinuity, meaning a common factor in the numerator and denominator cancels out.
- Placeholder for the coordinates of the hole: \( ([\text{Enter x-coordinate}], [\text{Enter y-coordinate}]) \) or "No hole".
Each characteristic requires evaluation based on the properties of the rational function given.
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