Answer :
To determine the domain and range of the function \(C(m) = 0.05m + 20\), let's analyze each component of this mathematical model step by step.
### Domain:
The domain of a function tells us the set of all possible input values. In our scenario, the input \(m\) represents the number of minutes Haley uses her cell phone in a month.
1. Since \(m\) represents minutes, it cannot be negative. Thus, the minimum value for \(m\) is \(0\).
2. There is no upper limit to the number of minutes that could potentially be used, so \(m\) can increase indefinitely.
This means the domain is all non-negative integers. Mathematically, we express this as:
[tex]\[ m \in [0, \infty) \][/tex]
### Range:
The range of the function is the set of all possible output values. In this case, the output \(C(m)\) represents the total cost of Haley's cell phone bill.
1. When \(m = 0\) (no minutes used), the cost \(C(0)\) is:
[tex]\[ C(0) = 0.05 \times 0 + 20 = 20 \][/tex]
Therefore, the minimum cost Haley will pay in a month is $20.
2. As \(m\) (the number of minutes) increases, the term \(0.05m\) increases, which means the total cost \(C(m)\) increases without any upper bound.
Hence, the range starts from 20 and extends to infinity. Mathematically, we express this as:
[tex]\[ C(m) \in [20, \infty) \][/tex]
### Final Answer:
Combining these analyses, we conclude:
- The domain of the function \(C(m) = 0.05m + 20\) is:
[tex]\[ m \in [0, \infty) \][/tex]
- The range of the function is:
[tex]\[ C(m) \in [20, \infty) \][/tex]
Thus, the complete answer is that the domain is [tex]\(m \in [0, \infty)\)[/tex] and the range is [tex]\(C(m) \in [20, \infty)\)[/tex].
### Domain:
The domain of a function tells us the set of all possible input values. In our scenario, the input \(m\) represents the number of minutes Haley uses her cell phone in a month.
1. Since \(m\) represents minutes, it cannot be negative. Thus, the minimum value for \(m\) is \(0\).
2. There is no upper limit to the number of minutes that could potentially be used, so \(m\) can increase indefinitely.
This means the domain is all non-negative integers. Mathematically, we express this as:
[tex]\[ m \in [0, \infty) \][/tex]
### Range:
The range of the function is the set of all possible output values. In this case, the output \(C(m)\) represents the total cost of Haley's cell phone bill.
1. When \(m = 0\) (no minutes used), the cost \(C(0)\) is:
[tex]\[ C(0) = 0.05 \times 0 + 20 = 20 \][/tex]
Therefore, the minimum cost Haley will pay in a month is $20.
2. As \(m\) (the number of minutes) increases, the term \(0.05m\) increases, which means the total cost \(C(m)\) increases without any upper bound.
Hence, the range starts from 20 and extends to infinity. Mathematically, we express this as:
[tex]\[ C(m) \in [20, \infty) \][/tex]
### Final Answer:
Combining these analyses, we conclude:
- The domain of the function \(C(m) = 0.05m + 20\) is:
[tex]\[ m \in [0, \infty) \][/tex]
- The range of the function is:
[tex]\[ C(m) \in [20, \infty) \][/tex]
Thus, the complete answer is that the domain is [tex]\(m \in [0, \infty)\)[/tex] and the range is [tex]\(C(m) \in [20, \infty)\)[/tex].
