Using asymptotic notation, consider two functions f(x) = 2n2 + 10n + 75 and g(x) = 100n + 300.
Is f(x) = O(g(x)) or f(x) = Ω (g(x))? State the values of constants n0 and C in support of your
answer. Also create a graph in Excel to demonstrate working.

Let's analyze the given functions and use asymptotic notation to determine whether \( f(x) = O(g(x)) \) or \( f(x) = \Omega(g(x)) \).

The functions are:
- \( f(x) = 2n^2 + 10n + 75 \)
- \( g(x) = 100n + 300 \)

### 1. **Determining if \( f(x) = O(g(x)) \):**

For \( f(x) = O(g(x)) \), we need to show that:
\[
f(x) \leq C \cdot g(x)
\]
for some constant \( C \) and for sufficiently large \( n \) (i.e., \( n \geq n_0 \)).

1.1. **Divide the dominant terms of \( f(x) \) and \( g(x) \):**

The dominant term of \( f(x) \) is \( 2n^2 \), and the dominant term of \( g(x) \) is \( 100n \).

As \( n \to \infty \), the quadratic term \( 2n^2 \) grows faster than the linear term \( 100n \). Therefore, \( f(x) \) does **not** grow asymptotically slower than \( g(x) \), so we cannot have \( f(x) = O(g(x)) \). Thus, \( f(x) \) is not \( O(g(x)) \).

### 2. **Determining if \( f(x) = \Omega(g(x)) \):**

For \( f(x) = \Omega(g(x)) \), we need to show that:
\[
f(x) \geq C \cdot g(x)
\]
for some constant \( C \) and for sufficiently large \( n \).

2.1. **Find the ratio of dominant terms**:  
Again, the dominant term of \( f(x) \) is \( 2n^2 \), and for \( g(x) \), it's \( 100n \). For sufficiently large \( n \), we need to check if the quadratic term \( 2n^2 \) grows at least as fast as some multiple of \( 100n \).

\[
\frac{f(x)}{g(x)} = \frac{2n^2 + 10n + 75}{100n + 300}
\]

For large \( n \), this ratio is approximately:

\[
\frac{2n^2}{100n} = \frac{2n}{100} = 0.02n
\]

Since \( 0.02n \) grows without bound as \( n \to \infty \), we conclude that \( f(x) \geq C \cdot g(x) \) for some constant \( C \), and thus \( f(x) = \Omega(g(x)) \).

### 3. **Finding \( n_0 \) and \( C \):**

We now need to find specific values of \( C \) and \( n_0 \) such that \( f(x) \geq C \cdot g(x) \) for all \( n \geq n_0 \).

Start with:
\[
2n^2 + 10n + 75 \geq C(100n + 300)
\]
For large \( n \), we focus on the dominant terms, so approximately:
\[
2n^2 \geq C \cdot 100n
\]
\[
n \geq \frac{C \cdot 100}{2}
\]
This simplifies to:
\[
n \geq 50C
\]

If we choose \( C = 0.02 \), then \( n_0 = 50 \). This means that for \( n \geq 50 \), \( f(x) \geq 0.02 \cdot g(x) \).

### 4. **Conclusion:**
Thus, \( f(x) = \Omega(g(x)) \) with constants \( C = 0.02 \) and \( n_0 = 50 \).

### 5. **Graph in Excel:**

I will now create a graph comparing \( f(x) \) and \( g(x) \) in Excel to visually demonstrate their growth rates. Please wait while I generate this graph.

The graph comparing \( f(x) = 2n^2 + 10n + 75 \) and \( g(x) = 100n + 300 \) has been created in Excel. You can download the file using the link below:

[Download the Excel file](https://file.aiquickdraw.com/chatgpt/1fc23869-ec4c-45c4-ac61-efccb1f35771//mnt/data/f_vs_g_graph.xlsx)

This graph will help visually demonstrate the growth rates of the two functions.

### Questions to explore:
1. What are the implications of \( f(x) = \Omega(g(x)) \) in algorithm analysis?
2. Can you find another pair of functions where \( f(x) = O(g(x)) \) but not \( \Omega(g(x)) \)?
3. How does Big-O notation relate to the efficiency of an algorithm?
4. What would the values of \( C \) and \( n_0 \) be if the functions had different coefficients?
5. Can you determine \( \Theta \) notation for these functions?

**Tip:** When working with asymptotic notations, focus on the dominant terms as they dictate the behavior for large \( n \).

Using asymptotic notation, consider two functions f(x) = 2n^2 + 10n + 75 and g(x) = 100n + 300. Is f(x) = O(g(x)) or f(x) = Ω(g(x))? State the values of constants n0 and C in support of your answer. Also create a graph in Excel to demonstrate working.

Explore whether f(x) = 2n² + 10n + 75 is Ω(g(x) = 100n + 300) using asymptotic notation. This analysis covers key concepts in algorithm complexity with a step-by-step solution and graph demonstration, suitable for undergraduate students.

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