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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Question
![**State the Domain for the Function**
The following problem asks you to determine the domain for the function given by:
\[ f(x) = \log_6(x - 5) + 4 \]
To determine the domain of the function, analyze the arguments within the logarithmic function and understand the constraints that must be satisfied.
### Options:
- \((-\infty, \infty)\)
- \((0, \infty)\)
- \((5, \infty)\)
- \((-\infty, 5)\)
**Explanation:**
In the given function \( \log_6(x - 5) \), the argument of the logarithm \( (x - 5) \) must be greater than zero since the logarithm is only defined for positive numbers.
Therefore, solving the inequality:
\[ x - 5 > 0 \]
\[ x > 5 \]
Thus, the domain of the function is \( (5, \infty) \).
You should select the third option:
- \((5, \infty)\)
This option represents the correct domain of the function \( f(x) = \log_6(x - 5) + 4 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f1ba715-f989-45d9-9496-7979ba76ee2a%2F12b7f4cf-9f6d-4d4a-b480-8774226913ae%2Fm48ev9a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**State the Domain for the Function**
The following problem asks you to determine the domain for the function given by:
\[ f(x) = \log_6(x - 5) + 4 \]
To determine the domain of the function, analyze the arguments within the logarithmic function and understand the constraints that must be satisfied.
### Options:
- \((-\infty, \infty)\)
- \((0, \infty)\)
- \((5, \infty)\)
- \((-\infty, 5)\)
**Explanation:**
In the given function \( \log_6(x - 5) \), the argument of the logarithm \( (x - 5) \) must be greater than zero since the logarithm is only defined for positive numbers.
Therefore, solving the inequality:
\[ x - 5 > 0 \]
\[ x > 5 \]
Thus, the domain of the function is \( (5, \infty) \).
You should select the third option:
- \((5, \infty)\)
This option represents the correct domain of the function \( f(x) = \log_6(x - 5) + 4 \).
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