Question

at 5:00 p.m., antonio turned on the oven. while the oven preheated, the temperature in the oven increased from 72°f to 400°f over a 10-minute period. the oven remained at 400°f for 45 minutes until antonio turned it off. it took 60 minutes for the temperature in the oven to cool, returning to 72°f at 6:55 p.m. which statement best explains whether or not the temperature in the oven is a function of the time?

• it is a function because at any given time the oven was exactly one temperature.
• it is a function because the oven temperature was the same at different times.
• it is not a function because at any given time the oven was exactly one temperature.
• it is not a function because the oven temperature was the same at different times.

Explanation:

Define the input and output variables

Using the Function Definition knowledge point

• Input variable (independent): Time \(t\) elapsed since 5:00 p.m.
• Output variable (dependent): Temperature \(T\) of the oven in \(^\circ\text{F}\).

Apply the definition of a function

Using the Function Definition knowledge point

• A relation is a function if each input value maps to exactly one output value.
• At any specific moment in time, the oven can only have one unique temperature.
• Therefore, the temperature is a function of time.

Evaluate the given options

Using the Function Definition knowledge point

• Option 1: True. Each time corresponds to exactly one temperature.
• Option 2: False. Having the same temperature at different times does not violate the function definition.
• Option 3: False. This describes a relation that is a function, but the reasoning is incorrect.
• Option 4: False. Having the same temperature at different times is allowed in a function.

Answer:

• (A) It is a function because at any given time the oven was exactly one temperature. (Correct answer)
• (B) It is a function because the oven temperature was the same at different times.
• (C) It is not a function because at any given time the oven was exactly one temperature.
• (D) It is not a function because the oven temperature was the same at different times.