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To find the profit function \( P(x) = R(x) - C(x) \), we need to subtract the cost function \( C(x) \) from the revenue function \( R(x) \).

Given:
\[
R(x) = -x^2 + 1800x
\]
\[
C(x) = 600x + 100
\]

Now, calculate the profit function \( P(x) = R(x) - C(x) \):

\[
P(x) = \left( -x^2 + 1800x \right) - \left( 600x + 100 \right)
\]

Simplify the expression:

\[
P(x) = -x^2 + 1800x - 600x - 100
\]

\[
P(x) = -x^2 + (1800x - 600x) - 100
\]

\[
P(x) = -x^2 + 1200x - 100
\]

So, the profit function is:

\[
P(x) = -x^2 + 1200x - 100
\]

The correct option is: \(\boxed{-x^2 + 1200x - 100}\)

Would you like more details on this process or any specific explanation?

Here are 5 related questions:
1. How does the shape of the profit function's graph look based on the quadratic term?
2. What happens to the profit as \(x\) (quantity) increases or decreases?
3. What are the implications of a negative quadratic term in the revenue function?
4. How can we find the maximum profit from this function?
5. How do fixed and variable costs affect the profit function?

**Tip:** When subtracting functions, always distribute the negative sign to ensure each term is correctly subtracted.

The revenue of a company is represented by R(x) and their costs are represented by C(x). What is the profit function R(x) - C(x)? Given R(x) = -x^2 + 1800x and C(x) = 600x + 100.

Learn how to find the profit function by subtracting the cost function from the revenue function. This problem covers quadratic expressions and is suitable for students in Grades 9-12.

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