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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
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→Y and ACX. The direct image of A under f is the set f(A) = {y €Y | 3a € A such that f(a) = y}. Remark. Another way to define the direct image of A under f is as the set ƒ(A) = {f(a) | a € A}. Problem 2. Do the following for the function f: R²R given by f(x, y) = x² + y². (a) For A = {(-1,0), (0, 1), (1,0), (2, 1)}, list the elements of f(A). (b) What is f(R²)? Is it all of R or something else? Problem 3. Let f: X→Y, and A and B be subsets of X. Prove the following statements. (a) If ACB, then f(A) ≤ f(B). (b) f(ANB) ≤ f(A) nf(B).
Transcribed Image Text:Definition. Let f: X→Y and ACX. The direct image of A under f is the set f(A) = {y €Y | 3a € A such that f(a) = y}. Remark. Another way to define the direct image of A under f is as the set ƒ(A) = {f(a) | a € A}. Problem 2. Do the following for the function f: R²R given by f(x, y) = x² + y². (a) For A = {(-1,0), (0, 1), (1,0), (2, 1)}, list the elements of f(A). (b) What is f(R²)? Is it all of R or something else? Problem 3. Let f: X→Y, and A and B be subsets of X. Prove the following statements. (a) If ACB, then f(A) ≤ f(B). (b) f(ANB) ≤ f(A) nf(B).
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