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The domain of f(e) = tan O - 2 is: All real numbers except for odd integers All real numbers except for odd multiples of All real numbers except for even integers All real numbers except for even multiples of 21

The domain of f(e) = tan O - 2 is: All real numbers except for odd integers All real numbers except for odd multiples of All real numbers except for even integers All real numbers except for even multiples of 21

Calculus: Early Transcendentals
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 the Function **Problem Statement:** The domain of \[ f(\theta) = \tan\left(\frac{1}{2}\pi\theta\right) - 2 \] is: **Options:** 1. All real numbers except for odd integers 2. All real numbers except for odd multiples of \(\frac{1}{2}\) 3. All real numbers except for even integers 4. All real numbers except for even multiples of \(\frac{1}{2}\) **Explanation:** To determine the domain of the given function, we need to identify values of \( \theta \) for which the function is undefined. - The tangent function, \(\tan(x)\), is undefined for: \[ x = \frac{\pi}{2} + k\pi \quad \text{where } k \text{ is any integer} \] In our function, \( x = \frac{1}{2}\pi\theta \). - Therefore, \[ \frac{1}{2}\pi\theta = \frac{\pi}{2} + k\pi \] - Solving for \( \theta \): \[ \theta = 1 + 2k \quad \text{where } k \text{ is any integer} \] Thus, \( \theta \) is undefined for odd integers. **Correct Answer:** - \(\quad\) \(\bigcirc\) All real numbers except for odd integers
Transcribed Image Text:### Determining the Domain of the Function **Problem Statement:** The domain of \[ f(\theta) = \tan\left(\frac{1}{2}\pi\theta\right) - 2 \] is: **Options:** 1. All real numbers except for odd integers 2. All real numbers except for odd multiples of \(\frac{1}{2}\) 3. All real numbers except for even integers 4. All real numbers except for even multiples of \(\frac{1}{2}\) **Explanation:** To determine the domain of the given function, we need to identify values of \( \theta \) for which the function is undefined. - The tangent function, \(\tan(x)\), is undefined for: \[ x = \frac{\pi}{2} + k\pi \quad \text{where } k \text{ is any integer} \] In our function, \( x = \frac{1}{2}\pi\theta \). - Therefore, \[ \frac{1}{2}\pi\theta = \frac{\pi}{2} + k\pi \] - Solving for \( \theta \): \[ \theta = 1 + 2k \quad \text{where } k \text{ is any integer} \] Thus, \( \theta \) is undefined for odd integers. **Correct Answer:** - \(\quad\) \(\bigcirc\) All real numbers except for odd integers
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