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Obtain the Lesson I.I.IE Resource Page, which is a set of four function machines. Your team's job is to use a specified input to get a particular output by putting those machines in order so that one machine's output becomes the next machine's input.

As you work, discuss what you know about the kind of output each machine produces to help you arrange the machines in an appropriate order.

The four functions are:
[tex]\[
\begin{array}{ll}
-2x + 34 & \frac{-x}{3} - 10 \\
-|3x| & (x - 2)^2
\end{array}
\][/tex]

a. In what order should you stack the machines so that when 15 is dropped into the first machine, and all four machines have had their effect, the last machine's output is -6?

b. When the initial input is 8, in what order should the machines be arranged so that the final output is 2?



Answer :

GinnyAnswer
Alright! Let's tackle this problem step-by-step.

### Step-by-Step Solution

#### Given Functions/Machines:
1. [tex]\( f_1(x) = -2x + 34 \)[/tex]
2. [tex]\( f_2(x) = \frac{-x}{3} - 10 \)[/tex]
3. [tex]\( f_3(x) = -|3x| \)[/tex]
4. [tex]\( f_4(x) = (x - 2)^2 \)[/tex]

We need to find the order of applying these functions (machines) to achieve a particular output from a given input.

### Part A
Initial Input = 15, Desired Output = -6

We need to determine the order of the machines such that applying all of them in sequence starting with an input of 15 results in -6.

According to the derived answer, the order is:
1. [tex]\( f_1 \)[/tex]
2. [tex]\( f_4 \)[/tex]
3. [tex]\( f_3 \)[/tex]
4. [tex]\( f_2 \)[/tex]

Let's verify this order step-by-step:

1. Function 1: [tex]\( f_1(x) = -2x + 34 \)[/tex]:
[tex]\[ f_1(15) = -2(15) + 34 = -30 + 34 = 4 \][/tex]

2. Function 4: [tex]\( f_4(x) = (x - 2)^2 \)[/tex]:
[tex]\[ f_4(4) = (4 - 2)^2 = 2^2 = 4 \][/tex]

3. Function 3: [tex]\( f_3(x) = -|3x| \)[/tex]:
[tex]\[ f_3(4) = -|3 \cdot 4| = -|12| = -12 \][/tex]

4. Function 2: [tex]\( f_2(x) = \frac{-x}{3} - 10 \)[/tex]:
[tex]\[ f_2(-12) = \frac{-(-12)}{3} - 10 = \frac{12}{3} - 10 = 4 - 10 = -6 \][/tex]

The final output is -6, which is the desired result.

### Part B
Initial Input = 8, Desired Output = 2

The order of functions derived to achieve this is:
1. [tex]\( f_3 \)[/tex]
2. [tex]\( f_2 \)[/tex]
3. [tex]\( f_4 \)[/tex]
4. [tex]\( f_1 \)[/tex]

Let's verify this order step-by-step:

1. Function 3: [tex]\( f_3(x) = -|3x| \)[/tex]:
[tex]\[ f_3(8) = -|3 \cdot 8| = -|24| = -24 \][/tex]

2. Function 2: [tex]\( f_2(x) = \frac{-x}{3} - 10 \)[/tex]:
[tex]\[ f_2(-24) = \frac{-(-24)}{3} - 10 = \frac{24}{3} - 10 = 8 - 10 = -2 \][/tex]

3. Function 4: [tex]\( f_4(x) = (x - 2)^2 \)[/tex]:
[tex]\[ f_4(-2) = (-2 - 2)^2 = (-4)^2 = 16 \][/tex]

4. Function 1: [tex]\( f_1(x) = -2x + 34 \)[/tex]:
[tex]\[ f_1(16) = -2(16) + 34 = -32 + 34 = 2 \][/tex]

The final output is 2, which is the desired result.

### Summary
- Part A: The order of the machines to convert an input of 15 to an output of -6 is: [tex]\( f_1, f_4, f_3, f_2 \)[/tex].
- Part B: The order of the machines to convert an input of 8 to an output of 2 is: [tex]\( f_3, f_2, f_4, f_1 \)[/tex].

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