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...
Related questions
Question
Find Domain and Range.
![The image shows the function:
\[ r(x) = \frac{6x - 6}{x + 2} \]
This is a rational function where the numerator is \(6x - 6\) and the denominator is \(x + 2\).
To simplify:
1. Factor the numerator: \(6x - 6 = 6(x - 1)\).
2. The function becomes \(\frac{6(x - 1)}{x + 2}\).
No further simplification is possible since the numerator and denominator do not have common factors that can be canceled.
Important points to consider when analyzing this function:
- **Vertical Asymptote**: The function has a vertical asymptote at \(x = -2\) where the denominator is zero, making the function undefined.
- **Horizontal Asymptote**: As \(x\) approaches infinity, the term \(6x/x = 6\), indicating a horizontal asymptote at \(y = 6\).
- **Domain**: The domain of the function is all real numbers except \(x = -2\).
Understanding these features helps in sketching the graph of the function and analyzing its behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24fae4ed-b1b1-4171-bba8-4d29349ae5e5%2F6e11bcd5-89c8-401e-b24a-dfae072b0dc3%2F1w6466_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows the function:
\[ r(x) = \frac{6x - 6}{x + 2} \]
This is a rational function where the numerator is \(6x - 6\) and the denominator is \(x + 2\).
To simplify:
1. Factor the numerator: \(6x - 6 = 6(x - 1)\).
2. The function becomes \(\frac{6(x - 1)}{x + 2}\).
No further simplification is possible since the numerator and denominator do not have common factors that can be canceled.
Important points to consider when analyzing this function:
- **Vertical Asymptote**: The function has a vertical asymptote at \(x = -2\) where the denominator is zero, making the function undefined.
- **Horizontal Asymptote**: As \(x\) approaches infinity, the term \(6x/x = 6\), indicating a horizontal asymptote at \(y = 6\).
- **Domain**: The domain of the function is all real numbers except \(x = -2\).
Understanding these features helps in sketching the graph of the function and analyzing its behavior.
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