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When calculating the rental cost for a car, you need to consider two main components: the daily rental charge and the mileage charge. The daily rental charge is a fixed cost based on the number of days you rent the vehicle. For example, if a car rental agency charges $49 per day and you rent for 2 days, this part of the cost would be calculated as \(49 \times 2 = 98\) dollars.
Additionally, the agency might charge per mile driven. Here, a charge of 25 cents per mile might apply. For 100 miles, this would translate into \(0.25 \times 100 = 25\) dollars.
To find the total cost, simply add the daily charge to the mileage charge, resulting in \(98 + 25 = 123\) dollars for this particular example. By understanding these components, you can more efficiently estimate your total rental charges in varied scenarios.

A function is often expressed in a single equation, but some situations require breaking it into pieces, hence the term "piecewise functions." In rental calculations, while the cost might seem simple, you can think of it in separate parts or 'pieces': the daily rental cost and the mileage cost.
Although the exercise did not explicitly call for a piecewise function, understanding this concept helps in organizing different costs within a single context. When different conditions or ranges result in different calculations (like different rental rates for weekend vs. weekday), piecewise functions are useful for tackling such scenarios.
The breakdown of cost into day and mileage charges illustrates how they function independently, converging into a full calculation.

Identifying variables is an essential skill in creating formulas that model real-world scenarios. When addressing rental cost calculations, it’s important to determine what each variable symbolizes. In our case, the formula has two primary variables:

• \(d\): Represents the number of days you rent the vehicle.
• \(m\): Refers to the number of miles driven during the rental period.

Defining each variable with understanding ensures that your formula is correct and applicable. It allows you to apply the function correctly in different situations, ensuring meaningful calculations any time values change.

Function notation is a way to represent mathematical relationships compactly. It’s especially beneficial for calculating costs based on changing inputs, something very common in rental cost problems. In this scenario, the total cost \(C\) is a function of both the number of days \(d\) and the number of miles \(m\). Thus, we can express it as: \[C(d, m) = 49d + 0.25m\]
This notation concisely communicates the relationship between the input variables and the resulting cost.
Using function notation makes recalculating cost easy as the values for \(d\) and \(m\) change. For example, to calculate the cost for driving from Dallas to Austin and back over a week, you simply substitute the values in: \(C(7, 500)\), meaning 7 days and 500 miles. The notation keeps the function clear, providing flexibility and simplicity in problem-solving.