express the following equation ????

Question

Express the following equation, using Ω Ο or Θ notation
2|x| ≤ | ((4x-5)(2x+5))/(3x^2+6) | ≤ 18|x|

Answer 1

Since (the complicated expression in the middle of the inequality) has both a lower bound, 2|x|, and an upper bound, 18|x|, this will most likely be big-Theta (Θ). The lower and upper bounds are each a constant multiple of the same function, |x|. Therefore:

| ((4x-5)(2x+5))/(3x^2+6) | = Θ(|x|)

It's worth noting that this function is not actually big-Theta of |x|, although it is big-Oh of |x|. But that's another issue.