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3 function machines, arranged so that the output of the first, becomes the input of the second, the output of the second becomes the input of the third. Rules for the 3 machines are: first: x, + 2, second, x squared, minus 4, x, third, square root of quantity, x, + 4. Mark claims to have created a sequence of three function machines that always gives him the same number he started with.

  1. Test his machines. Do you think he is right?

    Test his machines multiple times with different numbers.

  2. Be sure to test negative numbers. What happens for negative numbers?

    Again, test several different numbers.

  3. Mark wants to get his machines patented but has to prove that the set of machines will always do what he says it will, at least for positive numbers. Show Mark how to prove that his machines work for positive numbers by dropping in a variable (for example, ) and writing out each step the machines must take.

    machine 1:
    output 1:
    machine 2:
    output 2:

  4. Why do the negative numbers come out positive?

    Look at the steps that the variable went through in part (c). Pay careful attention to what happens when it passes through the third machine.

    Use the eTool below to test the machine.
    Click the link at right for the full version of the eTool: CCA2 6-26 HW eTool