Function Machines: Inputs, Outputs and Reversing

Picture a Saturday job that pays like this. Someone counts your hours, multiplies by your rate, then takes a fixed amount off at the end. A number lands in your account, and that number is the only part of it you ever actually see. So you know what came out. What you want to know is how many hours went in, and the only route to it is backwards through those same two steps. The tempting move is to undo them in the order you read them. Undo the multiply, then undo the take-off. Do it that way and the hours come out wrong, and wrong by an amount that still looks plausible enough that nobody goes back and checks it.

Start with the machine running the way it was built, which is forwards. A function machine is a picture of what happens to a number, in order. The number goes in on the left. It passes through one box, then a second box. Whatever leaves on the right is the output. Here is the machine we are working with. Multiply by four, then subtract five. Two boxes, and the order they sit in is part of the machine itself, not a detail about how someone chose to draw it. Feed six in on the left. Three outputs are on the screen: four, nineteen, or twenty four. Only one of them can come out of this machine. Take your pick. I'll wait. The answer is nineteen. Six times four is twenty four, and twenty four subtract five is nineteen. Twenty four is what you get if you stop after the first box. Four is what you get if you subtract the five first and multiply afterwards, and that one is worth holding on to, because it shows the order changes the answer even when you are going forwards.

Now turn the machine around, because the question you get asked is nearly always the backwards one. This time the output is nineteen, and the input is the number that has gone missing. Nothing about the machine has changed. You are simply walking through the same two boxes from the other end. Walking back, the first box you meet is the last box the number went through, and that is the subtract five. So the first thing you undo is the subtracting, not the multiplying. Getting dressed works exactly the same way. Socks first, then shoes. To undo that you take the shoes off first, because the shoes went on last. Last on, first off. So the reversed machine has the two boxes in the opposite order, and each box now does the opposite job. Subtract five becomes add five. Multiply by four becomes divide by four. Your turn, with that same machine. Multiply by four, then subtract five, and the output is nineteen. Write the reversed machine down first, both boxes flipped and swapped over, and only then find the input. Pause it there and work it out. I'll wait. The reversed machine reads add five, then divide by four, with nineteen going in on the left. Nineteen add five is twenty four. Twenty four divided by four is six. The input was six. Check it by running the original machine forwards. Six times four is twenty four, subtract five, nineteen. That is the output you were handed at the start, so six is right. Now watch what happens if you flip the operations but leave the order alone. You would divide nineteen by four first, which gives four point seven five, and then add five to reach nine point seven five. Push that back through the original machine and it falls apart. Nine point seven five times four is thirty nine, subtract five is thirty four, nowhere near nineteen. Flipping half the job gives you an answer that was never going to survive a check.

Examiners have written about this exact slip, and one report is worth hearing in the examiner's own words. It comes from a report on a paper where a function machine had to be run forwards in one part of the question and in reverse in the next part. Occasionally the function machine was misunderstood in part a, and the input of two was interpreted by changing the first operation box to plus two. Otherwise, this part was well answered, with just a few numerical errors made in otherwise correct methods. In part b, many candidates identified that it would be necessary to follow the function machine in reverse, but several made numerical errors. Use of divide by seven was more common than minus three. Some showed the correct complete inverse string, but then copied nine to the answer line rather than six. Of candidates who didn't identify the need to use inverse functions, a very common error was to use sixty three as the input rather than the output number. Take that last sentence first. Sixty three was the number that came out of the machine. Feeding it back in on the left runs the machine forwards all over again, when the whole point of the question was to run it backwards. Then there is the one where the working is right and the number written down is not. In a reversed machine, your input is the number that drops out of the very last box, so mark that number before you copy anything onto the answer line. And notice what happened in that first part. The machine got edited to fit the number, with an operation box changed to plus two. The boxes are fixed. The number is the only thing that is allowed to move.

Last on, first off. That short chant carries the whole method, so here is the method start to finish. Forwards, the number goes in on the left and meets the boxes in the order they are written. Backwards, you meet those boxes in the opposite order, and every one of them does the opposite job. Multiply becomes divide, and subtract becomes add. Draw the reversed machine as a machine before you calculate a single thing, and the order takes care of itself. Then feed your answer back through the original machine in the forwards direction. If the output that comes out matches the one you were given, you know your input is right.

Next in the chapter: Inverse and Composite Functions. That one chains two machines together, and gives your reversed machine its proper name.

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