Answer :
To solve this problem, we need to determine the reasonable domain and range for the function [tex]\(c(f) = 24f\)[/tex], where [tex]\(c(f)\)[/tex] is the number of cookies and [tex]\(f\)[/tex] is the number of cups of flour.
### Understanding the Function
1. Function Definition: The function [tex]\(c(f) = 24f\)[/tex] indicates that with each cup of flour, you can make 24 cookies.
### Determining the Domain
- The domain refers to the set of possible input values, which in this case, is the number of cups of flour [tex]\(f\)[/tex], and it makes sense to consider flour in non-negative values.
- Given that you have a finite amount of flour (between 0 and 8 cups as implied by practical considerations in the problem), the number of cups of flour, [tex]\(f\)[/tex], ranges from 0 to 8.
### Determining the Range
- The range is the set of possible output values, which is the number of cookies [tex]\(c(f)\)[/tex] that can be made.
- If [tex]\(f = 0\)[/tex], then [tex]\(c(f) = 24 \times 0 = 0\)[/tex] cookies (since no flour means no cookies).
- If [tex]\(f = 8\)[/tex], then [tex]\(c(f) = 24 \times 8 = 192\)[/tex] cookies.
- Therefore, the values [tex]\(c(f)\)[/tex] can take range from 0 to 192.
### Putting It Together
- Domain: The reasonable domain for [tex]\(f\)[/tex] is [tex]\(0 \leq f \leq 8\)[/tex].
- Range: The corresponding range for [tex]\(c(f)\)[/tex] is [tex]\(0 \leq c(f) \leq 192\)[/tex].
Thus, the correct choice that matches this reasoning is:
The domain is [tex]\(0 \leq f \leq 8\)[/tex]. The range is [tex]\(0 \leq c(f) \leq 192\)[/tex].
### Understanding the Function
1. Function Definition: The function [tex]\(c(f) = 24f\)[/tex] indicates that with each cup of flour, you can make 24 cookies.
### Determining the Domain
- The domain refers to the set of possible input values, which in this case, is the number of cups of flour [tex]\(f\)[/tex], and it makes sense to consider flour in non-negative values.
- Given that you have a finite amount of flour (between 0 and 8 cups as implied by practical considerations in the problem), the number of cups of flour, [tex]\(f\)[/tex], ranges from 0 to 8.
### Determining the Range
- The range is the set of possible output values, which is the number of cookies [tex]\(c(f)\)[/tex] that can be made.
- If [tex]\(f = 0\)[/tex], then [tex]\(c(f) = 24 \times 0 = 0\)[/tex] cookies (since no flour means no cookies).
- If [tex]\(f = 8\)[/tex], then [tex]\(c(f) = 24 \times 8 = 192\)[/tex] cookies.
- Therefore, the values [tex]\(c(f)\)[/tex] can take range from 0 to 192.
### Putting It Together
- Domain: The reasonable domain for [tex]\(f\)[/tex] is [tex]\(0 \leq f \leq 8\)[/tex].
- Range: The corresponding range for [tex]\(c(f)\)[/tex] is [tex]\(0 \leq c(f) \leq 192\)[/tex].
Thus, the correct choice that matches this reasoning is:
The domain is [tex]\(0 \leq f \leq 8\)[/tex]. The range is [tex]\(0 \leq c(f) \leq 192\)[/tex].
