I think I have a good grasp of what Big O, omega, and theta notation means and how to prove if a function is one of them. I don't understand how to prove a combination of them, like in the problem. Could somebody explain this to me?
Θ(n) + O(n^3) = O(n^3)
edit: typo, originally had said not equal
Actually, I think that Θ(n) + O(n^3) = O(n^3).
If you have a function f with constants k1 and k2 such that k1*n <= f(n) <= k2*n asymptotically (that's the Θ), and you have a function g with constant k3 such that g(n) <= k3*n^3 asymptotically (that's the left big-O), then there is a constant k4 such that f(n) + g(n) <= k4*n^3 asymptotically (the right big-O). Just take k4 = k2 + k3 and restrict yourself to n >= 1, because if n >= 1 then k2*n + k3*n^3 <= k2*n^3 + k3*n^3 = (k2 + k3)*n^3 = k4*n^3.
To show an inequality like the one you are asking for is simpler: then you just need to exhibit specific f and g that satify the bounds for the left hand side but where f(n) + g(n) does not satisfy the bounds for the right hand side.