determine if the functions are one to one y = √36-x², y = = = ² 2 y =√x+1 − 3 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Topic: Determining One-to-One Functions**
**Objective:** Determine if the following functions are one-to-one:
1. \( y = \sqrt{36 - x^2} \)
2. \( y = -\frac{1}{x+2} \)
3. \( y = \sqrt[3]{x+1} - 3 \)
**Explanation:**
**Function 1: \( y = \sqrt{36 - x^2} \)**
This is a semicircle equation with a radius of 6, centered at the origin on the x-axis. Since a semicircle does not pass the horizontal line test, it is not a one-to-one function.
**Function 2: \( y = -\frac{1}{x+2} \)**
This is a rational function that can be analyzed for one-to-oneness. It has a vertical asymptote at \( x = -2 \). The function is one-to-one because each input \( x \) gives a unique output, and it passes the horizontal line test.
**Function 3: \( y = \sqrt[3]{x+1} - 3 \)**
This function is a cubic root function modified by a vertical shift downward by 3 units. Cubic root functions are always one-to-one because they pass the horizontal line test, providing each input with a unique output.
**Conclusion:**
- Function 1 is not one-to-one.
- Function 2 is one-to-one.
- Function 3 is one-to-one.
Understanding whether a function is one-to-one is crucial for determining if it has an inverse that is also a function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09af1c01-6e60-4588-9390-2a0cbfc62f20%2Fac99442c-99e2-4de6-8406-36820f2d0307%2Fiopl37u3m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Determining One-to-One Functions**
**Objective:** Determine if the following functions are one-to-one:
1. \( y = \sqrt{36 - x^2} \)
2. \( y = -\frac{1}{x+2} \)
3. \( y = \sqrt[3]{x+1} - 3 \)
**Explanation:**
**Function 1: \( y = \sqrt{36 - x^2} \)**
This is a semicircle equation with a radius of 6, centered at the origin on the x-axis. Since a semicircle does not pass the horizontal line test, it is not a one-to-one function.
**Function 2: \( y = -\frac{1}{x+2} \)**
This is a rational function that can be analyzed for one-to-oneness. It has a vertical asymptote at \( x = -2 \). The function is one-to-one because each input \( x \) gives a unique output, and it passes the horizontal line test.
**Function 3: \( y = \sqrt[3]{x+1} - 3 \)**
This function is a cubic root function modified by a vertical shift downward by 3 units. Cubic root functions are always one-to-one because they pass the horizontal line test, providing each input with a unique output.
**Conclusion:**
- Function 1 is not one-to-one.
- Function 2 is one-to-one.
- Function 3 is one-to-one.
Understanding whether a function is one-to-one is crucial for determining if it has an inverse that is also a function.
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