To determine the correct range of values for [tex]\( x \)[/tex] that should be included in the graph of the bagel inventory function [tex]\( b(x) = 75 - 10x \)[/tex], let's analyze the context:
1. Initial Inventory and Rate of Sale:
- The coffee shop starts with 75 bagels.
- The shop sells bagels at a rate of 10 bagels per hour.
2. Understanding the Function [tex]\( b(x) \)[/tex]:
- [tex]\( b(x) = 75 - 10x \)[/tex] represents the bagel inventory [tex]\( x \)[/tex] hours after opening.
- When [tex]\( x = 0 \)[/tex] (shop opens), the inventory is [tex]\( b(0) = 75 \)[/tex].
- Each hour, the inventory decreases by 10 bagels.
3. Finding the Point When Inventory Depletes:
- To find when the inventory will reach zero:
[tex]\[
75 - 10x = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
75 = 10x
\][/tex]
[tex]\[
x = \frac{75}{10} = 7.5
\][/tex]
4. Determining Relevant Domain:
- The values of [tex]\( x \)[/tex] should start from 0 (when the shop opens) and go up to 7.5 hours (when the inventory is depleted).
Therefore, the correct range of [tex]\( x \)[/tex] values to include in the graph of the function for the relevant domain of the context is:
B. [tex]\( 0 \leq x \leq 7.5 \)[/tex]
