Let's analyze the given function \( C(h) = 25 + 50h \).
Independent Variable
The independent variable is \( h \), which represents the number of hours of service provided. The independent variable is the one that you can control or vary.
Dependent Variable
The dependent variable is \( C(h) \), which represents the total cost of the service. The dependent variable is the one that depends on or is affected by the independent variable.
Slope and Its Meaning
The slope of the function \( C(h) = 25 + 50h \) is 50. The slope represents the rate of change of the cost with respect to the number of hours of service. Specifically, it means that for each additional hour of service, the total cost increases by $50.
Y-Intercept and Its Meaning
The y-intercept of the function is 25, which means that when \( h = 0 \) (i.e., no hours of service are provided), the cost is $25. This represents the base fee charged for the service call itself, irrespective of the hours worked.
Domain
The domain of \( C(h) \) is the set of all possible values for \( h \). Since \( h \) represents hours of service, it can be 0 or more, but not negative. Thus, the domain is: \[ \text{Domain: } h \geq 0 \quad \text{or} \quad [0, \infty) \]
Range
The range of \( C(h) \) corresponds to the total cost based on possible values of \( h \). Since \( C(h) \) is calculated by \( C(h) = 25 + 50h \), and \( h \) can be 0 or greater:
- When \( h = 0 \), \( C(0) = 25 \)
- As \( h \) increases indefinitely, \( C(h) \) also increases indefinitely.
Thus, the range is: \[ \text{Range: } C(h) \geq 25 \quad \text{or} \quad [25, \infty) \]
In summary:
- Independent Variable: \( h \) (hours of service)
- Dependent Variable: \( C(h) \) (total cost)
- Slope: 50 (cost increases by $50 for each additional hour)
- Y-Intercept: 25 (base fee for the service call)
- Domain: \( [0, \infty) \)
- Range: \( [25, \infty) \)
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