Angelica is working with function machines. she has the two machines shown at right. she wants to put them in order so that the output of the first machine becomes the input of the second. her initial input is 6. a. in what order must she put the machines to get a final output of 5? b. is it possible for her to find an initial input that will get a final output of -5? if so, show how she could do that. if not, explain why not.

We can do simple trial and error for the first question. Trying the first machine first then the second gives us an output of -5(sqrt(6-5)=1, 1^2-6=-5), which we do not want. Trying the second machine first then the first (5^2-6=30, sqrt(30-5)=5) gives us 5, which is the correct answer.

We found one solution for the second question, so we know it's possible. Now we are looking to see if any others fit this. We can call the first machine's function f(x), and the second machine's function g(x), and we will just have to example f(g(x)) and g(f(x))

Now, we set f(g(x))=-5, and g(f(x))=-5 to get:

So we have -6 and 6 as two intial outputs that give us a final output of -5.

is only a line? i think so