Let's analyze the given function in the context provided:
Ethan uses the function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] to track the temperature of a substance over time.
The function [tex]\( f(x) \)[/tex] is specifically used to model the temperature of the substance as a function of [tex]\( x \)[/tex], where [tex]\( x \)[/tex] represents time in hours.
In mathematical terms, the domain of a function is the set of all possible input values (in this case, the values of [tex]\( x \)[/tex]) for which the function is defined. Here, [tex]\( x \)[/tex] represents the time over which Ethan is measuring the temperature.
Given that Ethan is tracking the temperature over a period of time (8 hours), the input for the function [tex]\( f(x) \)[/tex] is time. Therefore, the domain of Ethan's function is the set of all possible time values over which he is measuring the temperature.
Thus, the correct answer is:
Time
