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

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 .