Let's go through this problem step by step.
Part a: Identify the independent and dependent variables.
In the function [tex]\( n = -b + 30 \)[/tex]:
- The independent variable is the variable that we control or manipulate. Here, [tex]\( b \)[/tex] (the number of breakfast sandwiches) is the independent variable because it is the input value that determines the output.
- The dependent variable is the variable that depends on the independent variable. Here, [tex]\( n \)[/tex] (the number of eggs left) is the dependent variable because it changes depending on the value of [tex]\( b \)[/tex].
Thus:
- Independent variable: [tex]\( b \)[/tex] (number of breakfast sandwiches)
- Dependent variable: [tex]\( n \)[/tex] (number of eggs left)
Part b: Determine the range given the domain [tex]\( \{1, 2, 3, 4\} \)[/tex].
Now, we will substitute each value of [tex]\( b \)[/tex] from the domain into the function [tex]\( n = -b + 30 \)[/tex] to find the corresponding [tex]\( n \)[/tex] values, which will give us the range.
1. For [tex]\( b = 1 \)[/tex]:
[tex]\[
n = -1 + 30 = 29
\][/tex]
2. For [tex]\( b = 2 \)[/tex]:
[tex]\[
n = -2 + 30 = 28
\][/tex]
3. For [tex]\( b = 3 \)[/tex]:
[tex]\[
n = -3 + 30 = 27
\][/tex]
4. For [tex]\( b = 4 \)[/tex]:
[tex]\[
n = -4 + 30 = 26
\][/tex]
Thus, the range corresponding to the domain [tex]\( \{1, 2, 3, 4\} \)[/tex] is [tex]\( \{29, 28, 27, 26\} \)[/tex].
In conclusion:
- The range for the given domain is [tex]\( \{29, 28, 27, 26\} \)[/tex].
