To determine the relevant domain of the function that models the bagel inventory at a coffee shop, let's carefully analyze the problem step by step.
1. Understanding the Function:
- The coffee shop starts the day with 75 bagels.
- They sell an average of 10 bagels per hour.
- The function modeling the bagel inventory, [tex]\( b(x) \)[/tex], is given by:
[tex]\[
b(x) = 75 - 10x
\][/tex]
- Here, [tex]\( x \)[/tex] represents the number of hours after the shop opens, and [tex]\( b(x) \)[/tex] represents the number of bagels remaining.
2. Identify the Relevant Domain:
- The relevant domain is the range of [tex]\( x \)[/tex] values (hours) that makes sense within the context of the problem.
3. Bounding the Domain:
- The shop cannot sell bagels before it opens. Therefore, [tex]\( x \)[/tex] must be non-negative:
[tex]\[
x \geq 0
\][/tex]
- The shop sells 10 bagels every hour. To find out after how many hours all bagels will be sold out, set the inventory function to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
75 - 10x = 0 \implies 10x = 75 \implies x = \frac{75}{10} = 7.5
\][/tex]
- After 7.5 hours, the shop will have sold all 75 bagels. Thus, the upper bound for [tex]\( x \)[/tex] is 7.5 hours.
4. Conclusion:
- Therefore, the relevant domain of [tex]\( x \)[/tex], which reflects the period during which bagels are available for sale, is from 0 to 7.5 hours.
[tex]\[
0 \leq x \leq 7.5
\][/tex]
From the options provided:
- Option A: [tex]\( -\infty \leq x \leq \infty \)[/tex] is incorrect because negative hours and infinite hours do not make sense in the context of a shop's daily bagel sales.
- Option B: [tex]\( 0 \leq x \leq 75 \)[/tex] is incorrect because the shop runs out of bagels in just 7.5 hours, not 75 hours.
- Option C: [tex]\( 0 \leq x \leq \infty \)[/tex] is incorrect because the shop only sells bagels within the first 7.5 hours.
- Option D: [tex]\( 0 \leq x \leq 7.5 \)[/tex] is correct as it represents the time interval during which the shop has bagels to sell.
Hence, the correct answer is:
[tex]\[
\boxed{0 \leq x \leq 7.5}
\][/tex]
