f(x) = x 3 − 1000x^2 + x − 1
> x3 − 1000x^ 2
= (x − 1000)x^2
> (.9x)x^2
= .9x^3
Therefore, f(x) is Ω(x^3 ) with C = .9, k = 10, 000. Also, for all x > 0:
f(x) = x^3 − 1000x^2 + x − 1
< x^3 + 1000x^3 + x^3 + x^3
= 1002x^3
Therefore, f(x) is O(x^3 ) with C = 1002, k = 1.
