Problem 2. Do the following for the function f: R²R given by f(x, y) = 1² + y (a) For A = {(-1,0), (0, 1), (1,0), (2, 1)}, list the elements of f(4). (b) What is f(R²)? Is it all of R or something else?
Problem 2. Do the following for the function f: R²R given by f(x, y) = 1² + y (a) For A = {(-1,0), (0, 1), (1,0), (2, 1)}, list the elements of f(4). (b) What is f(R²)? Is it all of R or something else?
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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![**Definition.** Let \( f : X \to Y \) and \( A \subset X \). The **direct image** of \( A \) under \( f \) is the set
\[ f(A) = \{ y \in Y \mid \exists a \in A \text{ such that } f(a) = y \}. \]
**Remark.** Another way to define the direct image of \( A \) under \( f \) is as the set
\[ f(A) = \{ f(a) \mid a \in A \}. \]
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**Problem 2.** Do the following for the function \( f : \mathbb{R}^2 \to \mathbb{R} \) given by \( f(x, y) = x^2 + y^2 \).
(a) For \( A = \{(-1,0), (0,1), (1,0), (2,1)\} \), list the elements of \( f(A) \).
(b) What is \( f(\mathbb{R}^2) \)? Is it all of \( \mathbb{R} \) or something else?
---
**Problem 3.** Let \( f : X \to Y \), and \( A \) and \( B \) be subsets of \( X \). Prove the following statements.
(a) If \( A \subset B \), then \( f(A) \subset f(B) \).
(b) \( f(A \cap B) \subset f(A) \cap f(B) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f0128e8-6553-498c-afa0-dd14e88e258d%2Fef90ca62-1235-4c8a-b79b-3d7cebd8d709%2Fm93owyi_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition.** Let \( f : X \to Y \) and \( A \subset X \). The **direct image** of \( A \) under \( f \) is the set
\[ f(A) = \{ y \in Y \mid \exists a \in A \text{ such that } f(a) = y \}. \]
**Remark.** Another way to define the direct image of \( A \) under \( f \) is as the set
\[ f(A) = \{ f(a) \mid a \in A \}. \]
---
**Problem 2.** Do the following for the function \( f : \mathbb{R}^2 \to \mathbb{R} \) given by \( f(x, y) = x^2 + y^2 \).
(a) For \( A = \{(-1,0), (0,1), (1,0), (2,1)\} \), list the elements of \( f(A) \).
(b) What is \( f(\mathbb{R}^2) \)? Is it all of \( \mathbb{R} \) or something else?
---
**Problem 3.** Let \( f : X \to Y \), and \( A \) and \( B \) be subsets of \( X \). Prove the following statements.
(a) If \( A \subset B \), then \( f(A) \subset f(B) \).
(b) \( f(A \cap B) \subset f(A) \cap f(B) \).
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