A car rental company models the total cost (in dollars) as , where is the number of miles driven. In reality, mileage can be measured continuously, and the rental agreement allows up to 300 miles. Should the graph of versus be discrete points or continuous? Why?
- Discrete points, because miles must be whole numbers
- Continuous, because can be any real number from to (correct answer)
- Discrete points, because cost must be a whole number of dollars
- Continuous, for all real (no upper limit)
Explanation: This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. For C(d) = 45 + 0.30d, the variable d represents miles driven, which can be measured continuously (you can drive 47.3 miles, 150.82 miles, etc.). Modern odometers measure to tenths of a mile or better. The rental agreement sets bounds: 0 ≤ d ≤ 300. Since distance is continuous within these bounds, the graph should be a continuous line segment. Choice B correctly identifies the graph as continuous because d can be any real number from 0 to 300, reflecting that mileage is a continuous measurement. Choice A incorrectly assumes miles must be whole numbers (they don't), C confuses the output (cost) with the input (miles), and D ignores the 300-mile upper limit. Discrete vs continuous quick-check: ask 'can there be in-between values?' If you're modeling number of students in a classroom, going from 20 to 21 students happens instantly (no 20.5 students exist)—discrete! If you're modeling temperature change from 20°C to 21°C, every value in between occurs—continuous! Context tells you: countable/distinct objects → discrete (dots on graph), measurable/varying quantities → continuous (line/curve on graph).