Growth of functions. Using the definitions of Θ, Ο, and Ω show that:
a. 5 − 2 = Θ( )
b. 2 + 6 = O( )
c. 3 = Ω()
Required definitions -
1. “f(x) is O(g(x))”, if there are positive constants C and k such that
|f(x)| ≤ C |g(x)| for all x>k
2. "f(x) is Ω(g(x))", if there are positive constants C and k such that
|f(x)| ≥ C |g(x)| for all x>k
3. "f(x) is Θ(g(x))", if f(x) is both O(g(x)) and Ω(g(x)).
a. 5 - 2
Find for O(.)
5 - 2 = 3 = f(x)
Take g(x) = 1, C = 4
=> 3 ≤ 4
=> 5 - 2 = O(1)
Find for Ω(.)
Take g(x) = 1, C = 1
=> 3 ≥ 1
=> 5 - 2 = Ω(1)
=> 5 - 2 = Θ(1)
b. 2 + 6
2 + 6 = 8 = f(x)
Take g(x) = 1, C = 8
=> 8 ≤ 8
=> 2 + 6 = O(1)
c. 3
3 = f(x)
Take g(x) = 1, C = 1
=> 3 ≥ 1
=> 3 = Ω(1)
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