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11. Suppose f NN satisfies the recurrence relation f(n) if f(n) is even 3f(n)1 if f(n) is odd f(n1) 2 Note that with the initial condition f(0) 1, the values of the function are: f(1) 4 f(2)2. f(3) 1. f(4) 4 and so on, the images cycling through those three numbers. Thus f is NOT injective (and also certainly not surjective). Might it be under other initial conditions? 3 If f satisfies the initial condition f(0) 5, is finjective? Explain why or give a specific example of two elements from the domain with the same image. b. If f satisfies the initial condition f(0) 3, is f injective? Explain why or give a specific example of two elements from the domain with the same image. c. If f satisfies the initial condition f(0) 27, then it turns out that f(105) 10 and no two numbers less than 105 have the same image. Could f be injective? Explain. d. Prove that no matter what initial condition you choose, the function cannot be surjective.

11. Suppose f NN satisfies the recurrence relation f(n) if f(n) is even 3f(n)1 if f(n) is odd f(n1) 2 Note that with the initial condition f(0) 1, the values of the function are: f(1) 4 f(2)2. f(3) 1. f(4) 4 and so on, the images cycling through those three numbers. Thus f is NOT injective (and also certainly not surjective). Might it be under other initial conditions? 3 If f satisfies the initial condition f(0) 5, is finjective? Explain why or give a specific example of two elements from the domain with the same image. b. If f satisfies the initial condition f(0) 3, is f injective? Explain why or give a specific example of two elements from the domain with the same image. c. If f satisfies the initial condition f(0) 27, then it turns out that f(105) 10 and no two numbers less than 105 have the same image. Could f be injective? Explain. d. Prove that no matter what initial condition you choose, the function cannot be surjective.

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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11. Suppose f NN satisfies the recurrence relation f(n) if f(n) is even 3f(n)1 if f(n) is odd f(n1) 2 Note that with the initial condition f(0) 1, the values of the function are: f(1) 4 f(2)2. f(3) 1. f(4) 4 and so on, the images cycling through those three numbers. Thus f is NOT injective (and also certainly not surjective). Might it be under other initial conditions? 3 If f satisfies the initial condition f(0) 5, is finjective? Explain why or give a specific example of two elements from the domain with the same image. b. If f satisfies the initial condition f(0) 3, is f injective? Explain why or give a specific example of two elements from the domain with the same image. c. If f satisfies the initial condition f(0) 27, then it turns out that f(105) 10 and no two numbers less than 105 have the same image. Could f be injective? Explain. d. Prove that no matter what initial condition you choose, the function cannot be surjective.
Transcribed Image Text:11. Suppose f NN satisfies the recurrence relation f(n) if f(n) is even 3f(n)1 if f(n) is odd f(n1) 2 Note that with the initial condition f(0) 1, the values of the function are: f(1) 4 f(2)2. f(3) 1. f(4) 4 and so on, the images cycling through those three numbers. Thus f is NOT injective (and also certainly not surjective). Might it be under other initial conditions? 3 If f satisfies the initial condition f(0) 5, is finjective? Explain why or give a specific example of two elements from the domain with the same image. b. If f satisfies the initial condition f(0) 3, is f injective? Explain why or give a specific example of two elements from the domain with the same image. c. If f satisfies the initial condition f(0) 27, then it turns out that f(105) 10 and no two numbers less than 105 have the same image. Could f be injective? Explain. d. Prove that no matter what initial condition you choose, the function cannot be surjective.
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