## Answer :

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]