Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
Graph the following rational functions using transformations, state the domain, and all asymptotes
![### Function Analysis
**Problem 7.**
The function \( f(x) \) is defined as follows:
\[ f(x) = \frac{2x + 1}{x - 1} \]
#### Detailed Explanation:
- **Numerator:** The numerator of this rational function is \( 2x + 1 \).
- **Denominator:** The denominator is \( x - 1 \).
#### Key Considerations:
- **Domain:** The function is undefined where the denominator is zero. Here, the denominator \( x - 1 = 0 \) implies \( x = 1 \) is excluded from the domain.
- **Vertical Asymptote:** At \( x = 1 \), the function has a vertical asymptote since the function approaches infinity or negative infinity.
- **Behavior and Graphing:** While a specific graph is not provided, one can expect the rational function graph to have a hyperbolic shape with different behavior approaching the asymptote and across the domain. The exact behavior around the asymptote can be further analyzed using limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc48b0c6c-7fd3-4076-beff-88977fdedc32%2Fea8821fc-0ce7-4d2f-8737-82600dcfb98c%2Fmqmxhd_processed.png&w=3840&q=75)
Transcribed Image Text:### Function Analysis
**Problem 7.**
The function \( f(x) \) is defined as follows:
\[ f(x) = \frac{2x + 1}{x - 1} \]
#### Detailed Explanation:
- **Numerator:** The numerator of this rational function is \( 2x + 1 \).
- **Denominator:** The denominator is \( x - 1 \).
#### Key Considerations:
- **Domain:** The function is undefined where the denominator is zero. Here, the denominator \( x - 1 = 0 \) implies \( x = 1 \) is excluded from the domain.
- **Vertical Asymptote:** At \( x = 1 \), the function has a vertical asymptote since the function approaches infinity or negative infinity.
- **Behavior and Graphing:** While a specific graph is not provided, one can expect the rational function graph to have a hyperbolic shape with different behavior approaching the asymptote and across the domain. The exact behavior around the asymptote can be further analyzed using limits.
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