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Determine whether each of the functions 2n+l and 22n is

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 19E Chapter 3.2

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 19E

Determine whether each of the functions 2n+l and 22n is O(2n).

Step-by-Step Solution:
Step 1 of 3

Solution:

Step-1:

 In this problem we need to determine whether each of the functions and is .

 Note:

 Let us consider f and g are functions from the set of integers to the set of real numbers.

 The estimate value can be said that f(x) is O(g(x)) if there are constants C and k such that

 , where  C > 0 and  x> k.The constants C and k are called the witnesses to the relationship.

  The definition of f(x) is O(g(x)) says that f(x) grows slower than some fixed multiple of g(x) as x grows without bound.

Step-2:

Consider,

Now  we have to show that

         

           , since .

  Therefore , , where n > 1.

By using the above note , it is clear that C = 2, k = 1 and

Therefore , C = 2and k = 1 are witnesses for

Step-3:

Consider,

Now  we have to show that

         

           , since .

For every power of  , the term in p(n) has a power twice in value.

So,  

  Step-4:

Therefore , C = 2and k = 1 are witnesses for , and .

Step 2 of 3

Chapter 3.2, Problem 19E is Solved
Step 3 of 3

Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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Determine whether each of the functions 2n+l and 22n is