To determine the domain of the function [tex]\( M(x) = -40x + 720 \)[/tex] given the context of the problem, we need to consider the following factors:
1. Contextual Understanding:
- [tex]\( x \)[/tex] represents the number of months.
- [tex]\( M(x) \)[/tex] represents the remaining budget in dollars.
2. Constraints on [tex]\( x \)[/tex]:
- The number of months [tex]\( x \)[/tex] must be non-negative, so [tex]\( x \geq 0 \)[/tex].
- The business has a finite budget of [tex]$720, which means that after a certain number of months, the budget will be depleted.
3. Finding the Maximum \( x \):
- To find when the entire budget is spent, we set \( M(x) = 0 \):
\[
0 = -40x + 720
\]
Solving for \( x \):
\[
40x = 720
\]
\[
x = \frac{720}{40} = 18
\]
4. Interpreting the Result:
- This means after 18 months, the entire $[/tex]720 budget will be spent, and [tex]\( M(x) \)[/tex] will be zero.
- Therefore, [tex]\( x \)[/tex] can range from 0 months (when the campaign starts) to a maximum of 18 months (when the budget is completely used up).
Given these calculations, the domain of the function [tex]\( M(x) \)[/tex] is the set of possible values for [tex]\( x \)[/tex] considering the contextual constraints on the problem.
Thus, the domain is [tex]\([0, 18]\)[/tex], as the number of months [tex]\( x \)[/tex] must be between 0 and 18.
