Final answer:
The order of the functions should be: k, f, h, g. Starting with k(x)=2, taking its square root with f(x), scaling that with h(x) to reach the input of 2 for g(x), which finally returns 0.
Step-by-step explanation:
The student is looking to determine the order for stacking four functions, so that an initial input of 0 results in a final output of 0 after passing through all functions. The given functions are f(x) = √x, g(x) = -(x - 2)^2, h(x) = 2x - 7, and k(x) = 2. The order in which to stack the machines represents a sequence of function compositions that need to reverse or "undo" the operations of the others.
To solve this, we need to work backwards from an output of 0, looking for the correct arrangement of functions. Starting with k(x) = 2, which is a constant function, none of the other functions will change this value, indicating it's not a function that we can easily "undo" with the remaining functions. However, we want to reach a 0 by the end of the sequence, so we should consider g(x) = -(x - 2)^2 to be at the end since it can yield 0 when its input is 2. Now, looking at h(x) = 2x - 7, if we input 4.5, the output would be 2, leading us to consider f(x) = √x next. Since the square root of a number is undone by squaring that number, we recognize that squaring 2 gets us back to 4.
So, working forwards, we start with k(x) to produce 2, then apply f(x) to get the square root of 2, followed by h(x) which gives us 4.5 when inputting the square root of 2, and finally, g(x) can turn the 4.5 into 0. Thus, the correct stacking order to get a final result of 0 would be k, f, h, g.