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Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 3.2 - Problem 19e
Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 3.2 - Problem 19e

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

ISBN: 9780073383095 37

## Solution for problem 19E Chapter 3.2

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

Step 3 of 3

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