Task The purpose of this exploration is to help you develop your intuition about how common functions compare as they approach infinity. While there are many functions that we could compare, the ones that you need to compare as part of this exploration are the following: Exponential (e.g. er) Factorial (e.g. z!) Logarithmic (e.g. In(x)) . Polynomial (e.g. x - 3x2 + 100) In addition to developing your intuition about some common functions in this unit, the ideas in this exploration are very important in computer science. Mathematicians and computer scientists analyze algorithms to see if a computer has the capacity to solve them or not. We will not say more about this now, but if you are interested in the practical application of analyzing the end behavior of a given function, then look up the traveling salesman problem. It is a good introduction to this branch of mathematics. Defining the Race The first question that we need to answer is how should we define a function as the winner of a race to infinity? Before going on, take a few minutes and think about how you would define the winner. In this lesson, we will say that a function "wins" the race if at some point on the a-axis the function has the greatest y-value and no other function never passes it as a moves to infinity. Keep in mind that under different circumstances, we may be more interested in finding functions that grow slower. It isn't always the case that the fastest is best. Your answer will contain two parts: Part 1: Pick a Function Now that we have a common way to define the winner of the race, it is time for you to create the function that you think will win. There is only one rule: your function has to be either exponential, factorial, logarithmic, or polynomial. You can't mix functions (e.g. In a!"). The first part of this assignment is to create a function that you feel is faster than the other ones. You will want to make sure that you compare your function with ones from each of the other families of functions (i.e. exponential, factorial, logarithmic, and polynomial). Remember that you will need to justify your conclusion mathematically. Part 2: Critique the Following IdeasPart 2: Critique the Following Ideas Now that you have authored a function you think will win, critique each of the following potential student responses. Response 1 I have chosen e" because the function multiplies by e each time, but a! multiplies by smaller numbers each time. Response 2 I have chosen In (a) because it is the first function to get to infinity to the right. Response 3 I have chosen ! because I compared their graphs on Desmos. Even though the other functions started the fastest, ! passed the others after co. I couldn't figure out how to derive ! so I couldn't see that one for this function, but er was the winner in this situation