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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 44E
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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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