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,...
Related questions
Question
![**Graph the Following Function:**
\[ f(x) = \log_3 (2 - x) \]
**State the Asymptote:**
**State the Domain:**
**State the Range:**
**State the End Behavior:**
---
**Explanation of the Function:**
To graph the logarithmic function \( f(x) = \log_3 (2 - x) \), we need to understand its properties.
1. **Asymptote:**
- The vertical asymptote occurs where the argument of the logarithm equals zero, i.e., \( 2 - x = 0 \).
- Solving this, we get \( x = 2 \).
- Therefore, the vertical asymptote is at \( x = 2 \).
2. **Domain:**
- The domain of the function is determined by the argument of the logarithm being greater than zero.
- \( 2 - x > 0 \rightarrow x < 2 \).
- Therefore, the domain of the function is \( (-\infty, 2) \).
3. **Range:**
- The range of a logarithmic function is all real numbers.
- Therefore, the range of \( f(x) = \log_3 (2 - x) \) is \( (-\infty, \infty) \).
4. **End Behavior:**
- As \( x \to 2^- \) (approaching 2 from the left), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F488b60ff-9f36-4743-8e7c-aaf54d5d0a5e%2Fe31e4c3c-9fce-4234-aa07-d2cd715abcfa%2Ftzitxex_processed.png&w=3840&q=75)
Transcribed Image Text:**Graph the Following Function:**
\[ f(x) = \log_3 (2 - x) \]
**State the Asymptote:**
**State the Domain:**
**State the Range:**
**State the End Behavior:**
---
**Explanation of the Function:**
To graph the logarithmic function \( f(x) = \log_3 (2 - x) \), we need to understand its properties.
1. **Asymptote:**
- The vertical asymptote occurs where the argument of the logarithm equals zero, i.e., \( 2 - x = 0 \).
- Solving this, we get \( x = 2 \).
- Therefore, the vertical asymptote is at \( x = 2 \).
2. **Domain:**
- The domain of the function is determined by the argument of the logarithm being greater than zero.
- \( 2 - x > 0 \rightarrow x < 2 \).
- Therefore, the domain of the function is \( (-\infty, 2) \).
3. **Range:**
- The range of a logarithmic function is all real numbers.
- Therefore, the range of \( f(x) = \log_3 (2 - x) \) is \( (-\infty, \infty) \).
4. **End Behavior:**
- As \( x \to 2^- \) (approaching 2 from the left), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 1 images

Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you

Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON

Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON

Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON

Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press

College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education