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i. Prove that gi : Z → Z; g1 (x) = x – 4 is both one-to-one and onto. ii. Prove that g2 : R → R; g2(x) = |x| + x is neither one-to-one nor onto.

i. Prove that gi : Z → Z; g1 (x) = x – 4 is both one-to-one and onto. ii. Prove that g2 : R → R; g2(x) = |x| + x is neither one-to-one nor onto.

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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I need help with this proof question for my homework. Thanks!!

4. Working with functions. In this question, we will explore various properties of functions. You may want to review the basic definitions and terminology introduced on pages 15–16 of the course notes. Then, read the following definitions carefully. Definition: A function f : A → B is one-to-one iff no two elements of A have the same image. Symbol- ically, Va1, a2 E A, f(a1) = f(a2) → a1 = a2. (3) Definition: A function f : A → B is onto iff every element of B is the image of at least one element from A. Symbolically, E В, За € А, f (a) — . (4) Definition: For all functions f : A → B and g : B → C, their composition is the function gof : A → C defined by: Va e A, (go f)(a) = g(f(a)). (5) (a) i. Prove that gi : Z → Z; g1(x) = x – 4 is both one-to-one and onto. ii. Prove that g2 : R → R; 92(x) = |x| + x is neither one-to-one nor onto.
Transcribed Image Text:4. Working with functions. In this question, we will explore various properties of functions. You may want to review the basic definitions and terminology introduced on pages 15–16 of the course notes. Then, read the following definitions carefully. Definition: A function f : A → B is one-to-one iff no two elements of A have the same image. Symbol- ically, Va1, a2 E A, f(a1) = f(a2) → a1 = a2. (3) Definition: A function f : A → B is onto iff every element of B is the image of at least one element from A. Symbolically, E В, За € А, f (a) — . (4) Definition: For all functions f : A → B and g : B → C, their composition is the function gof : A → C defined by: Va e A, (go f)(a) = g(f(a)). (5) (a) i. Prove that gi : Z → Z; g1(x) = x – 4 is both one-to-one and onto. ii. Prove that g2 : R → R; 92(x) = |x| + x is neither one-to-one nor onto.
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