The data is modeled by the exponential function f(x) = 400(0.6)^x, where 400 represents the initial amount and 0.6 represents the decay factor for each time interval.
Step-by-step explanation:
Modeling data with an exponential function involves writing a function of the form f(x) = Abx, where A is the initial amount, b is the base of the exponential and represents the growth factor, and x is the exponent representing time or number of intervals.
Given the data points f(x)=400 when x=0, f(x)=240 when x=1, and f(x)=144 when x=2, we can observe that each time x increases by 1, the function decreases to 60% of its previous value, suggesting a decay factor of 0.6. Therefore, our base b is 0.6.
We can calculate the value of A by using the initial condition, where x is 0. Since 400 = Ab0, and b0 is 1, it follows that A = 400.
The exponential function representing this data set is therefore: f(x) = 400(0.6)x.