# Does big $\mathcal{O}$ imply $\Theta$

If we have a function $$f(x) = 6x^4 - 2x^3 + 5$$ and that function is $$\mathcal{O}(x^4)$$. Does that mean that it will also be $$\Omega(x^4)$$ and consequently $$\Theta(x^4)$$?

Presumably you're talking about $$x \to \infty$$. Then it is true that $$f(x) = \mathcal O(x^4)$$ and that $$f(x) = \Omega(x^4)$$, so $$f(x) = \Theta(x^4)$$. However, it's not always true that a function that is $$\mathcal O(x^4)$$ is also $$\Omega(x^4)$$. For example, $$x^3 = \mathcal O(x^4)$$, but not $$\Omega(x^4)$$.

In terms of $$x\to\infty$$, you have that a function $$f$$ is $$\Theta(g)$$ if it is $$\mathcal O(g)$$ and $$\Omega(g)$$ per definition, so you definitely have the inverse of your statement.

Coincidentally, the function you described is $$\Theta(x^4)$$ anyway. However, it is not always the case that $$f\in\mathcal O(g)$$ implies $$f\in\Theta(g)$$.

You can heuristically read $$\Theta$$ as "equals", $$\mathcal O$$ as less or equal than and $$\Omega$$ as greater or equal then.

As an example, you have e.g. that $$1\in\mathcal O(x^4)$$ but not $$1\in\Omega(x^4)$$, thus $$1\not\in\Theta(x^4)$$.

For your specific example, yes. In general: No.

Informally put: Let $$f: \mathbb{R^+} \mapsto \mathbb{R}^+$$ and $$g: \mathbb{Z} \mapsto \mathbb{R}^+$$ be two positive real-valued functions whose domain is the set of positive reals.

1. Then $$g$$ is $$O(f)$$ iff there are constant $$c>0$$ such that $$g(x) \le c \times f(x)$$ for all $$x$$.

2. Then $$g$$ is $$\Omega(f)$$ iff there is a constant $$c'>0$$ such that $$g(x) > c' \times f(x)$$ for all $$x$$.

3. Then $$g$$ is $$\theta(f)$$ iff BOTH $$g$$ is $$O(f)$$ AND $$g$$ is $$\Omega(f()$$.

4. If $$g$$ is $$O(f)$$ but not $$\Omega(f)$$ then $$g$$ is $$o(f)$$,

5. If $$g$$ is $$\Omega(f)$$ but not $$O(f)$$ then $$g$$ is $$\omega(f)$$.

Now letting $$f$$ be as in your question and $$g = x^4$$ indeed $$g$$ is both $$\Omega(f)$$ and $$g$$ is $$O(f)$$, so $$g$$ is indeed $$\theta(f)$$. But this follows becasue $$f$$ and $$g$$ are both polynomials of the same degree.