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Select the function that has a well-defined inverse. Of:Z Z f(x) = 2x – 5 Of:Z Z f(x) = [x/2] Of:Z→ Z* f(x) = |x| Of:Z Z f(x) = x + 4

Select the function that has a well-defined inverse. Of:Z Z f(x) = 2x – 5 Of:Z Z f(x) = [x/2] Of:Z→ Z* f(x) = |x| Of:Z Z f(x) = x + 4

Advanced Engineering Mathematics
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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**Problem Statement:** Select the function that has a well-defined inverse. **Options:** 1. \( f : \mathbb{Z} \to \mathbb{Z} \) \( f(x) = 2x - 5 \) 2. \( f : \mathbb{Z} \to \mathbb{Z} \) \( f(x) = \lfloor x/2 \rfloor \) 3. \( f : \mathbb{Z} \to \mathbb{Z}^+ \) \( f(x) = |x| \) 4. \( f : \mathbb{Z} \to \mathbb{Z} \) \( f(x) = x + 4 \) **Explanation:** 1. The function \( f(x) = 2x - 5 \) is a linear function with an inverse \( f^{-1}(x) = \frac{x + 5}{2} \), which is not well-defined over integers since division might not produce an integer. 2. The function \( f(x) = \lfloor x/2 \rfloor \) (floor function) is not one-to-one since different integers can yield the same floor value. 3. The function \( f(x) = |x| \) maps negative and positive values of \( x \) to the same positive value, therefore it is not one-to-one. 4. The function \( f(x) = x + 4 \) is a linear function where each input maps to a unique output. Its inverse is \( f^{-1}(x) = x - 4 \), which is well-defined over the integers. Therefore, the function \( f(x) = x + 4 \) is the correct choice as it has a well-defined inverse.
Transcribed Image Text:**Problem Statement:** Select the function that has a well-defined inverse. **Options:** 1. \( f : \mathbb{Z} \to \mathbb{Z} \) \( f(x) = 2x - 5 \) 2. \( f : \mathbb{Z} \to \mathbb{Z} \) \( f(x) = \lfloor x/2 \rfloor \) 3. \( f : \mathbb{Z} \to \mathbb{Z}^+ \) \( f(x) = |x| \) 4. \( f : \mathbb{Z} \to \mathbb{Z} \) \( f(x) = x + 4 \) **Explanation:** 1. The function \( f(x) = 2x - 5 \) is a linear function with an inverse \( f^{-1}(x) = \frac{x + 5}{2} \), which is not well-defined over integers since division might not produce an integer. 2. The function \( f(x) = \lfloor x/2 \rfloor \) (floor function) is not one-to-one since different integers can yield the same floor value. 3. The function \( f(x) = |x| \) maps negative and positive values of \( x \) to the same positive value, therefore it is not one-to-one. 4. The function \( f(x) = x + 4 \) is a linear function where each input maps to a unique output. Its inverse is \( f^{-1}(x) = x - 4 \), which is well-defined over the integers. Therefore, the function \( f(x) = x + 4 \) is the correct choice as it has a well-defined inverse.
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