Data Structures and Algorithms
with Object-Oriented Design Patterns in C#|
Consider the function f(n)=8n+128 shown in Figure . Clearly, f(n) is non-negative for all integers . We wish to show that . According to Definition , in order to show this we need to find an integer and a constant c>0 such that for all integers , .
It does not matter what the particular constants are--as long as they exist! For example, suppose we choose c=1. Then
Since (n+8)>0 for all values of , we conclude that . That is, .
So, we have that for c=1 and , for all integers . Hence, . Figure clearly shows that the function is greater than the function f(n)=8n+128 to the right of n=16.
Of course, there are many other values of c and that will do. For example, c=2 and will do, as will c=4 and . (See Figure ).
Figure: Showing that .