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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![### Determining the Domain of a Function
The domain of a function \( f(\theta) = \sec(\frac{4}{3} \pi \theta) - 2 \) is being sought. To find the correct domain, let's analyze the given multiple-choice options.
**Options:**
1. \( \bigcirc \) All real numbers except for even multiples of \( \frac{3}{4} \)
2. \( \bigcirc \) All real numbers except for odd multiples of \( \frac{3}{8} \)
3. \( \bigcirc \) All real numbers except for even multiples of \( \frac{3 \pi}{8} \)
4. \( \bigcirc \) All real numbers except for odd multiples of \( \frac{3 \pi}{8} \)
5. \( \bigcirc \) All real numbers except for even multiples of \( \frac{3}{8} \)
**Explanation:**
The function \( f(\theta) = \sec(\frac{4}{3} \pi \theta) - 2 \) involves the secant function, which is defined as \( \sec(x) = \frac{1}{\cos(x)} \). The secant function is undefined wherever the cosine function is zero. Therefore, we need to determine the values of \( \theta \) that make \( \cos(\frac{4}{3} \pi \theta) = 0 \).
The cosine function is zero at:
\[ \frac{4}{3} \pi \theta = \frac{\pi}{2} + n\pi \]
where \( n \) is any integer.
Solving for \( \theta \):
\[ \theta = \frac{3}{4} \left( \frac{\pi}{2} + n\pi \right) = \frac{3 \pi}{8} + \frac{3 \pi n}{4} \]
This implies that \( \theta \) will be in multiples of \( \frac{3 \pi}{8} \) plus an additional term dependent on \( n \).
Comparing with the options provided:
- Options involving odd and even multiples of \( \frac{3}{4} \) are incorrect because we're dealing with multiples of \( \frac{3}{8} \pi \).
- The correct option should involve excluding values where \( \theta \) are multiples of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f130582-8262-45ba-b1cd-6680cc0034f3%2Fdbcb5257-9969-4bb6-bef4-65661e3fb79f%2F1z83t55i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining the Domain of a Function
The domain of a function \( f(\theta) = \sec(\frac{4}{3} \pi \theta) - 2 \) is being sought. To find the correct domain, let's analyze the given multiple-choice options.
**Options:**
1. \( \bigcirc \) All real numbers except for even multiples of \( \frac{3}{4} \)
2. \( \bigcirc \) All real numbers except for odd multiples of \( \frac{3}{8} \)
3. \( \bigcirc \) All real numbers except for even multiples of \( \frac{3 \pi}{8} \)
4. \( \bigcirc \) All real numbers except for odd multiples of \( \frac{3 \pi}{8} \)
5. \( \bigcirc \) All real numbers except for even multiples of \( \frac{3}{8} \)
**Explanation:**
The function \( f(\theta) = \sec(\frac{4}{3} \pi \theta) - 2 \) involves the secant function, which is defined as \( \sec(x) = \frac{1}{\cos(x)} \). The secant function is undefined wherever the cosine function is zero. Therefore, we need to determine the values of \( \theta \) that make \( \cos(\frac{4}{3} \pi \theta) = 0 \).
The cosine function is zero at:
\[ \frac{4}{3} \pi \theta = \frac{\pi}{2} + n\pi \]
where \( n \) is any integer.
Solving for \( \theta \):
\[ \theta = \frac{3}{4} \left( \frac{\pi}{2} + n\pi \right) = \frac{3 \pi}{8} + \frac{3 \pi n}{4} \]
This implies that \( \theta \) will be in multiples of \( \frac{3 \pi}{8} \) plus an additional term dependent on \( n \).
Comparing with the options provided:
- Options involving odd and even multiples of \( \frac{3}{4} \) are incorrect because we're dealing with multiples of \( \frac{3}{8} \pi \).
- The correct option should involve excluding values where \( \theta \) are multiples of
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