I don’t know if you believe it or not, but I researched this subject a lot, asked AI a lot, watched a lot of videos, but believe me, I really didn’t understand anything. Maybe the question could be duplicated, but I wouldn’t write it here if I didn’t have to.

Why when O notation is shown, “f(x) = “x^3 + 2x^2 + 3” and “x^3 + 2x^2 + 3 ≤ x^3 + 2x^3 + 3x^3 and f(x) = x^2 + 2x^2 +3 < 6x^3″… I don’t understand why they write such things, so now if our function is “x^3 + 2x^2 + 3”, what does what is written on the other side of the inequality mean? So Have nested loops been added to the function in an unknown way?

As you know, when Omega is shown, this inequality is exactly the opposite.

and there are two different inequalities in theta, theta in the middle and at the edge “c1.g(x) <=f(x) <=c2.g(x)”.. Why did they write these? In the videos, they said something like the worst version of the function and the best version, but I didn’t understand it. What I know is that you write the function and that function remains as you wrote it, wouldn’t the loops be written into that function by itself afterwards?

Here are the things I understand:

For example: Big O Notation for f(x) = 5n^2 + 2n + 10 is very simply O(n^2). Because this value increases the most as the function goes to infinity asymptotically, and this is the worst case in which the function can be simplified and work. The Big O is shown this way. I understood that.

Big omega notation for f(x) = 5n^2 + 2n + 10
If the input n was an input that would be repeated only once, 5 + 2 + 10 = 17 for 5.1^2 + 2.1 + 10, that is, Omega(17). At best the algorithm has a time complexity of 17? But they would probably still call it n^2, but I don’t understand why they call it that, I can’t make sense of it logically.

Theta notation of the same function (if “Theta” is a notation that gives an average representation): If he divides O(n^2) by 2, then Theta becomes “n^2 / 2”? So in the average case, wouldn’t it work as half of the full case?

I know I spoke with very simple and overly plain logic, but as I said, this happened because I could not understand the subject. My math is limited to what I learned in high school. I did a lot of research, but this didn’t seem like something that could be solved with mathematics.

Maybe I need to give very simple examples to understand this, I don’t know. I’m just trying to pass an exam called “algorithm analysis”, I’m a student. I basically understood topics such as search and sorting algorithms, Binary search, bubble sort, huffman algorithm, as far as I was taught. But I don’t understand this part

I am a person with poor verbal abilities, this is really the case, I do not understand verbal expressions very much and even mathematics is a branch of science that is explained verbally and with word symbols. Maybe that’s why I worked so hard that I started thinking about things like this.

Thanks to everyone who answered, I hope the question doesn’t become a duplicate.