To solve this problem, let's take a careful look at the situation and the function provided.
The coffee shop starts the day with 75 bagels and sells an average of 10 bagels each hour. The function modeling the bagel inventory, [tex]\( b(x) \)[/tex], where [tex]\( x \)[/tex] represents the number of hours after opening, is given by:
[tex]\[ b(x) = 75 - 10x \][/tex]
We need to find the relevant domain for [tex]\( x \)[/tex] that makes sense within the context of this problem.
1. Initial Inventory: The coffee shop begins with 75 bagels when [tex]\( x = 0 \)[/tex] hours after opening. So initially, [tex]\( b(0) = 75 \)[/tex].
2. Selling Rate: The shop sells 10 bagels per hour. This implies that for every hour [tex]\( x \)[/tex] increases by 1, [tex]\( b(x) \)[/tex] decreases by 10.
3. Determining When Bagels Run Out: To find when the bagels will run out, we set the bagels inventory function [tex]\( b(x) \)[/tex] to 0 and solve for [tex]\( x \)[/tex]:
[tex]\[
75 - 10x = 0
\][/tex]
[tex]\[
10x = 75
\][/tex]
[tex]\[
x = 7.5
\][/tex]
This means that the coffee shop will run out of bagels after 7.5 hours.
4. Relevant Domain: The relevant domain is the interval from when the shop opens (0 hours) to when they run out of bagels (7.5 hours).
Therefore, the values for [tex]\( x \)[/tex] that should be included on the graph to represent the relevant domain are [tex]\( 0 \leq x \leq 7.5 \)[/tex].
Thus, the correct answer is:
A. [tex]\( 0 \leq x \leq 7.5 \)[/tex]
