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5. Determine whether each of these functions is a bijection from R to R. (a) f(x) = 2x – 10 %3D (b) f(x) = 4x² + 4 (c) f(x) = (x+ 1)/(x+2) (d) f(z) = r° +1

5. Determine whether each of these functions is a bijection from R to R. (a) f(x) = 2x – 10 %3D (b) f(x) = 4x² + 4 (c) f(x) = (x+ 1)/(x+2) (d) f(z) = r° +1

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 5:** Determine whether each of these functions is a bijection from \( \mathbb{R} \) to \( \mathbb{R} \). (a) \( f(x) = 2x - 10 \) (b) \( f(x) = 4x^2 + 4 \) (c) \( f(x) = \frac{x+1}{x+2} \) (d) \( f(x) = x^3 + 1 \)
Transcribed Image Text:**Problem 5:** Determine whether each of these functions is a bijection from \( \mathbb{R} \) to \( \mathbb{R} \). (a) \( f(x) = 2x - 10 \) (b) \( f(x) = 4x^2 + 4 \) (c) \( f(x) = \frac{x+1}{x+2} \) (d) \( f(x) = x^3 + 1 \)
**Question 4:** Provide examples of functions from ℕ (natural numbers) to ℕ that satisfy the following conditions: (a) **One-to-one but not onto:** A function \( f: \mathbb{N} \to \mathbb{N} \) is one-to-one if every element in the domain maps to a unique element in the codomain, but not all elements in the codomain are mapped to by the domain. Example: \( f(n) = 2n \). (b) **Onto but not one-to-one:** A function is onto if every element in the codomain has a pre-image in the domain, but it is not one-to-one if different elements in the domain map to the same element in the codomain. Example: \( f(n) = \lfloor n/2 \rfloor \). (c) **Both onto and one-to-one (but different from the identity function \( f(x) = x \)):** A function that is both one-to-one and onto, meaning it is bijective. Such a function maps every element of the domain to a unique element of the codomain with no repeated mappings. A non-identity example: \( f(n) = n + 1 \) for \( n \ge 2 \) and \( f(1) = 1 \). (d) **Neither one-to-one nor onto:** A function that fails to meet both criteria, meaning multiple elements in the domain map to the same element in the codomain and not all codomain elements are covered. Example: \( f(n) = 1 \) for all \( n \).
Transcribed Image Text:**Question 4:** Provide examples of functions from ℕ (natural numbers) to ℕ that satisfy the following conditions: (a) **One-to-one but not onto:** A function \( f: \mathbb{N} \to \mathbb{N} \) is one-to-one if every element in the domain maps to a unique element in the codomain, but not all elements in the codomain are mapped to by the domain. Example: \( f(n) = 2n \). (b) **Onto but not one-to-one:** A function is onto if every element in the codomain has a pre-image in the domain, but it is not one-to-one if different elements in the domain map to the same element in the codomain. Example: \( f(n) = \lfloor n/2 \rfloor \). (c) **Both onto and one-to-one (but different from the identity function \( f(x) = x \)):** A function that is both one-to-one and onto, meaning it is bijective. Such a function maps every element of the domain to a unique element of the codomain with no repeated mappings. A non-identity example: \( f(n) = n + 1 \) for \( n \ge 2 \) and \( f(1) = 1 \). (d) **Neither one-to-one nor onto:** A function that fails to meet both criteria, meaning multiple elements in the domain map to the same element in the codomain and not all codomain elements are covered. Example: \( f(n) = 1 \) for all \( n \).
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