To determine the range of the function [tex]\( c = 30.75u + 5.75 \)[/tex] for the number of uniforms [tex]\( u \)[/tex] within the range of at least 8 players but not more than 12 players, we need to find the total cost for both the minimum and maximum values of [tex]\( u \)[/tex].
1. Calculate the total cost for the minimum number of players (8):
- Using [tex]\( u = 8 \)[/tex]:
[tex]\[
c = 30.75(8) + 5.75
\][/tex]
[tex]\[
c = 246 + 5.75
\][/tex]
[tex]\[
c = 251.75
\][/tex]
So, the total cost is [tex]\( 251.75 \)[/tex] dollars when 8 uniforms are bought.
2. Calculate the total cost for the maximum number of players (12):
- Using [tex]\( u = 12 \)[/tex]:
[tex]\[
c = 30.75(12) + 5.75
\][/tex]
[tex]\[
c = 369 + 5.75
\][/tex]
[tex]\[
c = 374.75
\][/tex]
So, the total cost is [tex]\( 374.75 \)[/tex] dollars when 12 uniforms are bought.
Thus, the range of the function [tex]\( c = 30.75u + 5.75 \)[/tex] for [tex]\( u \)[/tex] ranging from 8 to 12 is from [tex]\( 251.75 \)[/tex] to [tex]\( 374.75 \)[/tex].
This means the range of the function for this situation is [tex]\( \boxed{251.75 \text{ to } 374.75} \)[/tex]. Therefore, the correct answer is:
[tex]\[
0 < c \leq 374.75
\][/tex]
However, given the options provided in the question, the best match is:
[tex]\[
\{251.75, 282.50, 313.25, 344.00, 374.75\}
\][/tex]
The correct option is [tex]\(\boxed{\{251.75,282.50,313.25,344,374.75\}}\)[/tex].
