Invent an interesting and original exponential function, f(t). Assume that f(t) represents the size of something t years from now. (g) If your answer to (c) was “growing,” how long will it take the size of the thing modeled by \( f \) to double? If your answer to (c) was “shrinking,” how long will it take the size of the thing modeled by \( f \) to be reduced to half its original size? **Problem 2: Exploring Exponential Functions**Invent an interesting and original exponential function, \( f(t) \). Assume that \( f(t) \) represents the size of something \( t \) years from now.**Questions:**(a) How much of the quantity is present now?(b) Draw a good graph of \( y = f(t) \).(c) Is \( f \) growing or shrinking?(d) What is the annual growth rate of \( f \)?(e) Express \( f(t) \) in the form \( f(t) = Ae^{kt} \). What is the continuous growth rate of \( f \)?(f) How much of the something modeled by \( f \) will there be 13 years from now?

Assume that f(t) represents the size of something t years from now. Then, as an exponential function we define f(t)=Aekt. Consider the time t for now as 0. So at t = 0, the function gives f(0)=Aek0=A Hence, the quantity present now is A. Obtain the graph of the function f(t)=Aekt as follows: If k is positive, then the function is growing and if k is negative, then the function is shrinking. Annual growth rate is f'(t)=Akekt. It is the continuous growth rate. If it is growing, to make it double 2A=Aekt⇒t=ln2k,k>0 and if it is shrinking, to make it half A2=Aekt⇒t=-ln2k, k<0 For inventing, we need to observe something like a radioactive material, population and so on.