Find the domain of y = cos ¹(2x).
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Problem 37:**
Find the domain of \( y = \cos^{-1}(2x) \).
**Solution:**
The function \( y = \cos^{-1}(x) \) is defined for \( x \) in the interval \([-1, 1]\). Therefore, for the given function \( y = \cos^{-1}(2x) \), the expression inside the inverse cosine, \( 2x \), must also be within the interval \([-1, 1]\).
**Steps to find the domain:**
1. **Set the expression within the valid range:**
\[
-1 \leq 2x \leq 1
\]
2. **Solve the inequalities:**
- Divide each part of the inequality by 2:
\[
-\frac{1}{2} \leq x \leq \frac{1}{2}
\]
Thus, the domain of \( y = \cos^{-1}(2x) \) is:
\[
x \in \left[-\frac{1}{2}, \frac{1}{2}\right]
\]
This means that the function is defined for all \( x \) within this interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05d85d42-88ea-4e71-ac38-898b8539ea7f%2F29249979-d97c-4db0-b587-54225d56042a%2F9n17ti_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 37:**
Find the domain of \( y = \cos^{-1}(2x) \).
**Solution:**
The function \( y = \cos^{-1}(x) \) is defined for \( x \) in the interval \([-1, 1]\). Therefore, for the given function \( y = \cos^{-1}(2x) \), the expression inside the inverse cosine, \( 2x \), must also be within the interval \([-1, 1]\).
**Steps to find the domain:**
1. **Set the expression within the valid range:**
\[
-1 \leq 2x \leq 1
\]
2. **Solve the inequalities:**
- Divide each part of the inequality by 2:
\[
-\frac{1}{2} \leq x \leq \frac{1}{2}
\]
Thus, the domain of \( y = \cos^{-1}(2x) \) is:
\[
x \in \left[-\frac{1}{2}, \frac{1}{2}\right]
\]
This means that the function is defined for all \( x \) within this interval.
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