This notation bounds a function to in constant factors. We say f(n) = Θ(g(n)) if there presents positive constants n_{0}, c_{1} and c_{2} such that to the right of n_{0} the value of f(n) always lies between c1g(n) and c2g(n), both inclusive. The given Figure gives an idea about function f(n) and g(n) where f(n) = Θ(g(n)) . We will say that the function g(n) is asymptotically tight bound for f(n).
Figure: Plot of f(n) = Θ(g(n))
For instance, let us show that the function f(n) = (1/3) n^{2} - 4n = Θ(n^{2}).
Now, we must determined three positive constants, c _{1}, c _{2} and n_{o} as
c_{1}n^{2} ≤1/3 n^{2} - 4n ≤ c_{2 }n^{2} for all n ≥ no
⇒ c_{1} ≤ 1/3- 4/n ≤ c_{2}
By choosing no = 1 and c_{2} ≥ 1/3 the right hand inequality holds true.
Likewise, by choosing n_{o} = 13, c_{1} ≤ 1/39, the right hand inequality holds true. Thus, for c1 = 1/39 , c_{2} = 1/3 and n_{o} ≥ 13, it follows that 1/3 n^{2} - 4n = Θ (n^{2}).
Surely, there are other alternative for c_{1}, c_{2} and no. Now we might illustrates that the function f(n) = 6n^{3} ≠ Θ (n_{2}).
In order to prove this, let us suppose that c_{3} and no exist such that 6n^{3} ≤ c_{3}n^{2} for n ≥ n_{o}, But this fails for adequately large n. Therefore 6n^{3} ≠ Θ (n^{2}).