The Greatest Output

Challenge#

Name:

Greatest Output Think of a function as a math machine with an input and an output. Suppose the function is A(x) = 3x + 1. That means if you put any number (x) into this function machine, the machine will multiply the number by three and add one. For example, what is the output if five is the input? Five times three plus one equals 16 for the output. That's written A(5) = 16. Suppose you join two function machines so that the output of the first one is connected to the input of the second one. Let's make the second function B(x) = x2. If you put two into the A machine, out comes seven. Then seven is the input to the B machine that squares it, and out comes 49. That's written B(A(2)) = 49. Here are five functions: A(x) = 4x - 3 B(x) = 3x2 C(x) = 7x + 1 D(x) = x2 E(x) = 2x + 7 If connected in alphabetical order with two as the input number. That would be written as E(D(C(B(A(2))))). The resulting output = 553,359. What arrangement of functions will produce the largest output with two as the input (each function used only once)? Use logical reasoning and algebra rather than just guess and check. Explain why you chose your particular arrangement. Extension Would your arrangement of functions necessarily be the best for all input numbers? Can you find other input numbers for which a different order of the functions would produce the largest output? For what input numbers would the order change? What would be the order of functions to produce the largest output for this input number?