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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![Find the domain
\( f(t) = \pi + \log_8 \left( \frac{t-1}{\sqrt{8-t}} \right) \)
### Explanation:
To determine the domain of the function \( f(t) = \pi + \log_8 \left( \frac{t-1}{\sqrt{8-t}} \right) \), consider the following conditions that must be met for the logarithm to be defined:
1. The argument of the logarithm must be positive:
\[ \frac{t-1}{\sqrt{8-t}} > 0 \]
2. The expression inside the square root must be non-negative:
\[ 8 - t > 0 \]
\[ t < 8 \]
3. \( t-1 \) must be positive to keep the entire fraction positive:
\[ t-1 > 0 \]
\[ t > 1 \]
Combining these conditions, the domain of \( f(t) \) is \( 1 < t < 8 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93b779eb-3fb4-4e8e-a3e5-cbc97ed5211a%2F9958f679-fcc8-4be1-97ea-29bb8efd92ab%2Fp39rpb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Find the domain
\( f(t) = \pi + \log_8 \left( \frac{t-1}{\sqrt{8-t}} \right) \)
### Explanation:
To determine the domain of the function \( f(t) = \pi + \log_8 \left( \frac{t-1}{\sqrt{8-t}} \right) \), consider the following conditions that must be met for the logarithm to be defined:
1. The argument of the logarithm must be positive:
\[ \frac{t-1}{\sqrt{8-t}} > 0 \]
2. The expression inside the square root must be non-negative:
\[ 8 - t > 0 \]
\[ t < 8 \]
3. \( t-1 \) must be positive to keep the entire fraction positive:
\[ t-1 > 0 \]
\[ t > 1 \]
Combining these conditions, the domain of \( f(t) \) is \( 1 < t < 8 \).
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