For each of the following functions, prove that the function is neither injective nor surjective. Then, show how you could restrict the domain and codomain - without changing the mapping rule – to make the function both injective and surjective. Your restricted domain and codomain should be as large as possible. When the domain and codomain are infinite to begin with, you should interpret "as large as possible" to mean that the restricted domain and
For each of the following functions, prove that the function is neither injective nor surjective. Then, show how you could restrict the domain and codomain - without changing the mapping rule – to make the function both injective and surjective. Your restricted domain and codomain should be as large as possible. When the domain and codomain are infinite to begin with, you should interpret "as large as possible" to mean that the restricted domain and
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![For each of the following functions, prove that the function is neither injective nor
surjective. Then, show how you could restrict the domain and codomain – without changing
the mapping rule – to make the function both injective and surjective. Your restricted domain
and codomain should be as large as possible. When the domain and codomain are infinite to
begin with, you should interpret "as large as possible" to mean that the restricted domain and
restricted codomain must also be infinite.
For example, the function f: R+R given by f(r) = r² is neither injective nor surjective, but
we can restrict the domain and codomain to define the function f : [0, 0) → [0, 0) given by
f(x) = 22, which is both injective and surjective.
%3D
(a) g: {a, b,
d} → {a, e, i, o, u} given by:
r g(x)
a
a
b
i
i
e
(b) h: W Z where W is the set of all finite subsets of Z, given by h(S) = |S].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc25e7241-f0c7-4aa0-a6bb-f43bb5726ed7%2Fc87d555a-0cc1-4a11-ae97-69643625fe7e%2Fa3mc4y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For each of the following functions, prove that the function is neither injective nor
surjective. Then, show how you could restrict the domain and codomain – without changing
the mapping rule – to make the function both injective and surjective. Your restricted domain
and codomain should be as large as possible. When the domain and codomain are infinite to
begin with, you should interpret "as large as possible" to mean that the restricted domain and
restricted codomain must also be infinite.
For example, the function f: R+R given by f(r) = r² is neither injective nor surjective, but
we can restrict the domain and codomain to define the function f : [0, 0) → [0, 0) given by
f(x) = 22, which is both injective and surjective.
%3D
(a) g: {a, b,
d} → {a, e, i, o, u} given by:
r g(x)
a
a
b
i
i
e
(b) h: W Z where W is the set of all finite subsets of Z, given by h(S) = |S].
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