Question
A cell phone plan cost $30 for each line
a. Does the situation or persona function? If so identify the independent and dependent variables
b. You can have a maximum of four lines on a plan. find the domain and range
a. Does the situation or persona function? If so identify the independent and dependent variables
b. You can have a maximum of four lines on a plan. find the domain and range
Answers
GPT-4o mini
Let's analyze the given cell phone plan scenario step-by-step:
### a. Functions Analysis
The cost of the cell phone plan is given by the equation:
\[ C = 30 \times n \]
where:
- \( C \) is the total cost of the plan,
- \( n \) is the number of lines.
#### Independent and Dependent Variables
- **Independent variable**: This is the number of lines (\( n \)), which you can choose or control.
- **Dependent variable**: This is the total cost (\( C \)), which depends on the number of lines you have.
Since there's a clear relationship where the cost changes based on the number of lines, this situation does indeed represent a function.
### b. Domain and Range
#### Domain
The domain represents the possible values for the independent variable, which is the number of lines (\( n \)). According to the problem, you can have a maximum of four lines, and given that you can also have zero lines, the domain can be expressed as:
\[ \text{Domain} = \{ 0, 1, 2, 3, 4 \} \]
In interval notation, this can be expressed as:
\[ \text{Domain} = [0, 4] \] (considering these are integer values)
#### Range
The range represents the possible values for the dependent variable, which is the total cost (\( C \)).
Calculating the cost for each value in the domain:
- If \( n = 0 \): \( C = 30 \times 0 = 0 \)
- If \( n = 1 \): \( C = 30 \times 1 = 30 \)
- If \( n = 2 \): \( C = 30 \times 2 = 60 \)
- If \( n = 3 \): \( C = 30 \times 3 = 90 \)
- If \( n = 4 \): \( C = 30 \times 4 = 120 \)
Thus the range of values for \( C \) is:
\[ \text{Range} = \{ 0, 30, 60, 90, 120 \} \]
In interval notation, this can be expressed as:
\[ \text{Range} = [0, 120] \] (again, considering these are specific cost amounts)
### Summary
- **Independent variable**: Number of lines (\( n \))
- **Dependent variable**: Total cost (\( C \))
- **Domain**: \( \{ 0, 1, 2, 3, 4 \} \)
- **Range**: \( \{ 0, 30, 60, 90, 120 \} \)
### a. Functions Analysis
The cost of the cell phone plan is given by the equation:
\[ C = 30 \times n \]
where:
- \( C \) is the total cost of the plan,
- \( n \) is the number of lines.
#### Independent and Dependent Variables
- **Independent variable**: This is the number of lines (\( n \)), which you can choose or control.
- **Dependent variable**: This is the total cost (\( C \)), which depends on the number of lines you have.
Since there's a clear relationship where the cost changes based on the number of lines, this situation does indeed represent a function.
### b. Domain and Range
#### Domain
The domain represents the possible values for the independent variable, which is the number of lines (\( n \)). According to the problem, you can have a maximum of four lines, and given that you can also have zero lines, the domain can be expressed as:
\[ \text{Domain} = \{ 0, 1, 2, 3, 4 \} \]
In interval notation, this can be expressed as:
\[ \text{Domain} = [0, 4] \] (considering these are integer values)
#### Range
The range represents the possible values for the dependent variable, which is the total cost (\( C \)).
Calculating the cost for each value in the domain:
- If \( n = 0 \): \( C = 30 \times 0 = 0 \)
- If \( n = 1 \): \( C = 30 \times 1 = 30 \)
- If \( n = 2 \): \( C = 30 \times 2 = 60 \)
- If \( n = 3 \): \( C = 30 \times 3 = 90 \)
- If \( n = 4 \): \( C = 30 \times 4 = 120 \)
Thus the range of values for \( C \) is:
\[ \text{Range} = \{ 0, 30, 60, 90, 120 \} \]
In interval notation, this can be expressed as:
\[ \text{Range} = [0, 120] \] (again, considering these are specific cost amounts)
### Summary
- **Independent variable**: Number of lines (\( n \))
- **Dependent variable**: Total cost (\( C \))
- **Domain**: \( \{ 0, 1, 2, 3, 4 \} \)
- **Range**: \( \{ 0, 30, 60, 90, 120 \} \)