To determine the practical domain of the function [tex]\( f(p) = 12p + 5 \)[/tex], we need to consider what values of [tex]\( p \)[/tex] are realistic in the context of the problem.
Here, [tex]\( p \)[/tex] represents the number of packages of rolls the caterer can order. Since each package contains 12 rolls, the number of packages [tex]\( p \)[/tex] must be a whole number (integer), and it cannot be zero or a negative number because the number of packages ordered must make sense in the context of ordering rolls.
Additionally, the caterer can order up to 9 packages of rolls. Therefore, [tex]\( p \)[/tex] must be a whole number between 1 and 9, inclusive.
The practical domain, in this case, is the set of all integer values between 1 and 9, inclusive. Thus, [tex]\( p \)[/tex] can be any integer from 1 to 9.
The practical domain of the function [tex]\( f(p) = 12p + 5 \)[/tex] is therefore:
[tex]\[ \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \][/tex]
This corresponds to the set of all integers from 1 to 9, inclusive, making the proper choice the option: "all integers from 1 to 9, inclusive".
