To determine the domain and range of the function [tex]\( C(m) = 0.05m + 20 \)[/tex], where [tex]\( m \)[/tex] represents the number of minutes used, we need to understand the relationship between [tex]\( m \)[/tex] and [tex]\( C(m) \)[/tex].
### Domain of the Function [tex]\( C(m) \)[/tex]
The domain of a function consists of all possible values that the independent variable (in this case, [tex]\( m \)[/tex]) can take.
1. Identify the Independent Variable: Here, [tex]\( m \)[/tex] represents the number of minutes used.
2. Consider Realistic Values for [tex]\( m \)[/tex]: Since [tex]\( m \)[/tex] represents the number of minutes used on a cell phone, it must be a non-negative integer.
Therefore, the domain of [tex]\( C(m) \)[/tex] is all non-negative integers. This can be written as:
[tex]\[ m \ge 0 \][/tex]
### Range of the Function [tex]\( C(m) \)[/tex]
The range of a function consists of all possible values that the dependent variable (in this case, [tex]\( C(m) \)[/tex]) can take.
1. Evaluate [tex]\( C(m) \)[/tex] at Its Minimum Value: Since [tex]\( m \ge 0 \)[/tex], the smallest value [tex]\( m \)[/tex] can be is 0.
[tex]\[
C(0) = 0.05 \cdot 0 + 20 = 20
\][/tex]
Therefore, the minimum value of [tex]\( C(m) \)[/tex] is 20.
2. Consider [tex]\( m \)[/tex] Increasing: As [tex]\( m \)[/tex] increases, [tex]\( C(m) \)[/tex] increases linearly. There is no upper bound for [tex]\( m \)[/tex], so [tex]\( C(m) \)[/tex] can be as large as desired.
Thus, the range of [tex]\( C(m) \)[/tex] starts from 20 and includes all values greater than or equal to 20. This can be written as:
[tex]\[ C(m) \ge 20 \][/tex]
### Summary
- Domain: The domain of [tex]\( C(m) \)[/tex] is [tex]\( m \ge 0 \)[/tex].
- Range: The range of [tex]\( C(m) \)[/tex] is [tex]\( C(m) \ge 20 \)[/tex].
So, the function [tex]\( C(m) \)[/tex] has:
- Domain: [tex]\( m \ge 0 \)[/tex]
- Range: [tex]\( C(m) \ge 20 \)[/tex], starting from the smallest value 20.0
These values summarize the constraints and behavior of Haley's cell phone bill function.
