Answer:
A, D, C, B
Explanation:
It can be convenient to work from a table of input and output values for the given functions. Such a table is shown in the first attachment.
Working from a table
The attached table shows the function output values for various inputs. The first row is for the first input of 15. Subsequent rows are for inputs that come from the outputs shown on previous rows.
The first input gives 4 possible outputs. Each of those gives 4 more, so 16 possible outputs by the 3rd machine, there are 64 possible outputs that could go into the 4th machine.
It became apparent that some outputs are way out of range for providing reasonable inputs, so these were not considered.
As it happened, we found that we could get an output of -6 after just three machines. Working through the required input/output sequence, it became apparent that one machine produced the same output as its input for one of the values in the chain. This simplified the process somewhat.
Sequence
Using the table, we determined the required sequence of operations is ...
15 —A— 4 —D— 4 —C— -12 —B— -6
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Additional comment
The overall function is ...
y = b(c(d(a(x))))
y = b(c(((-2x+34) -2)²)) = b(c(4(x -16)²)) = b(-|3(4(x-16)²)|) = b(-12(x -16)²)
y = -(-12(x -16)²/3) -10
y = 4(x -16)² -10 . . . . . . function of machines A, D, C, B in order
For x=15, this is y=4(15 -16)² -10 = 4 -10 = -6 . . . . . as required.
Note that (x -16)² is always positive, so the absolute value function does nothing to it.