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Exercise 4. Let p: R→ R be a polynomial function, that is, assume that p is given by an expression of the form p(x) = x" +an-12-¹+...+ao. a) Assume that n is an odd number, that is, assume that n is not divisible by 2. Show that p is surjective. b) Find an example of a polynomial p of degree three such that p is not injective. c) Assume that n is an even number, that is, assume that n is divisible by 2. Show that p is not surjective nor injective.

Exercise 4. Let p: R→ R be a polynomial function, that is, assume that p is given by an expression of the form p(x) = x" +an-12-¹+...+ao. a) Assume that n is an odd number, that is, assume that n is not divisible by 2. Show that p is surjective. b) Find an example of a polynomial p of degree three such that p is not injective. c) Assume that n is an even number, that is, assume that n is divisible by 2. Show that p is not surjective nor injective.

Calculus: Early Transcendentals
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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DEFINITION 3 (INJECTIVE FUNCTION). Let A, B C R be two subsets of the real numbers, and let f : A → B be a function. We say that f is injective (we might alternatively say that f is 1-to-1) if for all a₁, a2 € A such that f(a₁) = f(a₂) we have that a₁ = a2. DEFINITION 4 (SURJECTIVE FUNCTION). Let A, B C R be two subsets of the real numbers, and let f : A → B be a function. We say that f is surjective (we might alternatively say that f is onto) if for every b = B we can find a € A such that f(a) = b. Exercise 4. Let p: R → R be a polynomial function, that is, assume that p is given by an expression of the form p(x) = x² +an-1-¹+...+ao. a) Assume that n is an odd number, that is, assume that n is not divisible by 2. Show that p is surjective. b) Find an example of a polynomial p of degree three such that p is not injective. c) Assume that n is an even number, that is, assume that n is divisible by 2. Show that p is not surjective nor injective.
Transcribed Image Text:DEFINITION 3 (INJECTIVE FUNCTION). Let A, B C R be two subsets of the real numbers, and let f : A → B be a function. We say that f is injective (we might alternatively say that f is 1-to-1) if for all a₁, a2 € A such that f(a₁) = f(a₂) we have that a₁ = a2. DEFINITION 4 (SURJECTIVE FUNCTION). Let A, B C R be two subsets of the real numbers, and let f : A → B be a function. We say that f is surjective (we might alternatively say that f is onto) if for every b = B we can find a € A such that f(a) = b. Exercise 4. Let p: R → R be a polynomial function, that is, assume that p is given by an expression of the form p(x) = x² +an-1-¹+...+ao. a) Assume that n is an odd number, that is, assume that n is not divisible by 2. Show that p is surjective. b) Find an example of a polynomial p of degree three such that p is not injective. c) Assume that n is an even number, that is, assume that n is divisible by 2. Show that p is not surjective nor injective.
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