Domain and Range (Part 2) Cool Down: A Pot of Water The function
gives the temperature, in degrees Fahrenheit, of a pot of water on a stove
minutes after the stove is turned on. After 30 minutes, the pot is taken off the stove. The graph of the function is shown. temperature (degrees Fahrenheit) 300 250 200 150 100 50 0 5 10 15 20 25 30 time (minutes) 1. Is 250 in the range of function
? Explain how you know. 2. Describe the range of the function. 3. Does
have a solution? Explain how you know.
The Correct Answer and Explanation is:1
Let’s break down each part of the Cool Down: A Pot of Water question, assuming the graph shows a curve that starts at a lower temperature (e.g., room temperature), rises as the pot heats up, and levels off near or below boiling point, ending at 30 minutes when the pot is taken off the stove.
1. Is 250 in the range of the function? Explain how you know.
Yes, 250 is in the range of the function.
Explanation:
The range of a function consists of all possible output values (in this case, temperatures) that the function can produce over the given domain (0 to 30 minutes). From the graph, we can see that the temperature increases over time. If the temperature starts around room temperature (say 70–80°F) and gradually rises as the stove heats the pot, it will pass through all the intermediate temperatures up to its maximum value before 30 minutes. Since 250°F is less than the typical boiling point of water (212°F), it may not be reached in reality, but since the graph shows values going up to around 250 or even higher, we can conclude that 250 is included. If there’s a point on the graph where the temperature equals 250°F, or if the graph is continuous and passes through that value, then 250 is in the range.
2. Describe the range of the function.
The range is approximately from the initial temperature (about 70°F) up to the maximum temperature the water reaches (around 280°F).
Explanation:
The function models heating water on a stove, so it likely starts at room temperature and increases steadily as time progresses. The range includes all temperatures the water reaches from the start to the 30-minute mark. Since the graph is continuous and increasing, the range would include all the values between the starting temperature and the highest point on the graph. If, for instance, the graph starts at 70°F and peaks at 280°F by 30 minutes, then the range would be [70, 280] in degrees Fahrenheit.
3. Does the function have a solution for T(w) = 300? Explain how you know.
No, the function does not have a solution for T(w)=300T(w) = 300.
Explanation:
A solution exists only if the function reaches that temperature at some point within the domain (0 ≤ w ≤ 30). From the graph, the temperature increases over time but does not reach 300°F. If the graph levels off or ends before hitting 300°F, then there is no time ww for which T(w)=300T(w) = 300. Therefore, 300°F is not in the range, so the equation T(w)=300T(w) = 300 has no solution.
Summary:
- 250°F is in the range.
- The range is a continuous interval, e.g., [70, 280].
- There is no solution to T(w)=300T(w) = 300, because 300°F is never reached on the graph.
