Practice two-step function machines with this educational worksheet designed to enhance understanding of input-output relationships and inverse operations.

Let’s solve this step by step.

We have three sections: A, B, and C. Each has function machines with operations like “+2”, “×3”, etc. We need to find outputs (Section A), inputs (Section B), or missing values (Section C).

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Section A: Apply the operations to find the outputs

Each machine has two steps. For example, first box says:
Input → +2 → ×1 → Output

So for input 1:
1 + 2 = 3 → 3 × 1 = 3 → Output = 3
Input 2: 2 + 2 = 4 → 4 × 1 = 4
Input 3: 3 + 2 = 5 → 5 × 1 = 5
Input 4: 4 + 2 = 6 → 6 × 1 = 6

Second machine: Input → +1 → ×2 → Output
Input 1: 1 + 1 = 2 → 2 × 2 = 4
Input 2: 2 + 1 = 3 → 3 × 2 = 6
Input 3: 3 + 1 = 4 → 4 × 2 = 8
Input 4: 4 + 1 = 5 → 5 × 2 = 10

Third machine: Input → ×3 → -2 → Output
Input 1: 1 × 3 = 3 → 3 - 2 = 1
Input 2: 2 × 3 = 6 → 6 - 2 = 4
Input 4: 4 × 3 = 12 → 12 - 2 = 10
Input 6: 6 × 3 = 18 → 18 - 2 = 16

Fourth machine: Input → -2 → ×4 → Output
Input 3: 3 - 2 = 1 → 1 × 4 = 4
Input 5: 5 - 2 = 3 → 3 × 4 = 12
Input 7: 7 - 2 = 5 → 5 × 4 = 20
Input 9: 9 - 2 = 7 → 7 × 4 = 28

Fifth machine: Input → +2 → ×3 → Output
Input 8: 8 + 2 = 10 → 10 × 3 = 30
Input 10: 10 + 2 = 12 → 12 × 3 = 36
Input 16: 16 + 2 = 18 → 18 × 3 = 54
Input 28: 28 + 2 = 30 → 30 × 3 = 90

Sixth machine: Input → -1 → ÷5 → Output
Input 16: 16 - 1 = 15 → 15 ÷ 5 = 3
Input 21: 21 - 1 = 20 → 20 ÷ 5 = 4
Input 31: 31 - 1 = 30 → 30 ÷ 5 = 6
Input 46: 46 - 1 = 45 → 45 ÷ 5 = 9
Input 61: 61 - 1 = 60 → 60 ÷ 5 = 12

✔ Section A done.

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Section B: Apply inverse operations to find the inputs

This means we go backwards through the operations.

First machine: Input → +3 → ÷2 → Output
To reverse: Start from output, multiply by 2, then subtract 3.

Output 11: 11 × 2 = 22 → 22 - 3 = 19 → Input = 19
Output 17: 17 × 2 = 34 → 34 - 3 = 31
Output 23: 23 × 2 = 46 → 46 - 3 = 43
Output 35: 35 × 2 = 70 → 70 - 3 = 67

Second machine: Input → +2 → ×3 → Output
Reverse: Divide by 3, then subtract 2.

Output 18: 18 ÷ 3 = 6 → 6 - 2 = 4
Output 27: 27 ÷ 3 = 9 → 9 - 2 = 7
Output 42: 42 ÷ 3 = 14 → 14 - 2 = 12
Output 51: 51 ÷ 3 = 17 → 17 - 2 = 15

Third machine: Input → -1 → ÷4 → Output
Reverse: Multiply by 4, then add 1.

Output 3: 3 × 4 = 12 → 12 + 1 = 13
Output 8: 8 × 4 = 32 → 32 + 1 = 33
Output 11: 11 × 4 = 44 → 44 + 1 = 45

Fourth machine: Input → ÷5 → +3 → Output
Reverse: Subtract 3, then multiply by 5.

Output 8: 8 - 3 = 5 → 5 × 5 = 25
Output 11: 11 - 3 = 8 → 8 × 5 = 40
Output 15: 15 - 3 = 12 → 12 × 5 = 60

Fifth machine: Input → ×10 → -4 → Output
Reverse: Add 4, then divide by 10.

Output 44: 44 + 4 = 48 → 48 ÷ 10 = 4.8? Wait — that doesn’t look right. Let me check again.

Wait — maybe I misread. The machine is: Input → ×10 → -4 → Output

So if output is 44:
Let input be x → x × 10 - 4 = 44 → x × 10 = 48 → x = 4.8? But all other numbers are whole numbers. Maybe it's a typo? Or perhaps I should double-check.

Actually, looking back at the image description — in Section B, fifth machine: Input → ×10 → -4 → Output, and outputs given are 44, 234, 50, 5.

Let’s do:

For output 44:
x × 10 - 4 = 44 → x × 10 = 48 → x = 4.8 → Not integer. Hmm.

But wait — maybe the operation order is different? No, it says “×10” then “-4”.

Alternatively, perhaps the problem expects us to work with decimals? But let’s check others.

Output 234: x × 10 - 4 = 234 → x × 10 = 238 → x = 23.8
Output 50: x × 10 - 4 = 50 → x × 10 = 54 → x = 5.4
Output 5: x × 10 - 4 = 5 → x × 10 = 9 → x = 0.9

That seems odd. Maybe I made a mistake in reading the machine?

Wait — looking again at the original problem description: In Section B, fifth machine is written as:

“Input → ×10 → -4 → Output”

And outputs: 44, 234, 50, 5

Perhaps it’s meant to be solved as is, even if decimal. But let’s hold on — maybe there’s an error in my assumption.

Alternatively, perhaps the machine is “÷10” instead of “×10”? But no, the text says “×10”.

Wait — let me re-express:

If output = (input × 10) - 4

Then input = (output + 4) / 10

So:

44 → (44+4)/10 = 48/10 = 4.8
234 → 238/10 = 23.8
50 → 54/10 = 5.4
5 → 9/10 = 0.9

Okay, so they are decimals. Maybe that’s acceptable.

But let’s continue.

Sixth machine: Input → ×2 → -5 → Output
Reverse: Add 5, then divide by 2.

Output 41: 41 + 5 = 46 → 46 ÷ 2 = 23
Output 17: 17 + 5 = 22 → 22 ÷ 2 = 11
Output 2: 2 + 5 = 7 → 7 ÷ 2 = 3.5
Output -4: -4 + 5 = 1 → 1 ÷ 2 = 0.5

Again, decimals. Okay, we’ll go with that.

✔ Section B done.

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Section C: Fill in missing inputs and outputs

First machine: Input → +7 → -4 → Output

Given:
Input 2 → ? → Output ?
Output 31 → Input ?
Input 12 → ? → Output ?
Output 45 → Input ?

Step-by-step:

Machine: Input +7 then -4 → so overall: Input + 3

Because +7 -4 = +3

So Output = Input + 3

Therefore:

Input 2 → Output = 2 + 3 = 5
Output 31 → Input = 31 - 3 = 28
Input 12 → Output = 12 + 3 = 15
Output 45 → Input = 45 - 3 = 42

Second machine: Input → +5 → ÷3 → Output

Overall: (Input + 5) ÷ 3 = Output

Given:
Input 7 → Output = (7+5)/3 = 12/3 = 4
Output 13 → Input + 5 = 13 × 3 = 39 → Input = 39 - 5 = 34
Output 12 → Input + 5 = 12 × 3 = 36 → Input = 31
Output 20 → Input + 5 = 60 → Input = 55

Wait — let me write clearly:

For Input 7: (7 + 5) ÷ 3 = 12 ÷ 3 = 4 → Output = 4

For Output 13: Let input be x → (x + 5) ÷ 3 = 13 → x + 5 = 39 → x = 34

For Output 12: (x + 5) ÷ 3 = 12 → x + 5 = 36 → x = 31

For Output 20: (x + 5) ÷ 3 = 20 → x + 5 = 60 → x = 55

Third machine: Input → -2 → ×8 → Output

Overall: (Input - 2) × 8 = Output

Given:
Output 64 → (x - 2) × 8 = 64 → x - 2 = 8 → x = 10
Output 88 → (x - 2) × 8 = 88 → x - 2 = 11 → x = 13
Input 5 → (5 - 2) × 8 = 3 × 8 = 24
Input 11 → (11 - 2) × 8 = 9 × 8 = 72

✔ Section C done.

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Now, compiling all answers.

Since the user didn’t specify which part to answer, but the task is to solve the problem accurately, I will provide all filled values as per sections.

But note: The final answer should be concise. Since this is a worksheet with many blanks, perhaps list them section by section.

However, the instruction says: “Provide only the final correct answer — this should come AFTER all explanations and reasoning.”

So I will summarize the completed tables.

But to keep it clean, I'll present the answers in order as they appear.

Final Answer:

Section A Outputs (left to right, top to bottom):

First machine (+2, ×1): 3, 4, 5, 6
Second machine (+1, ×2): 4, 6, 8, 10
Third machine (×3, -2): 1, 4, 10, 16
Fourth machine (-2, ×4): 4, 12, 20, 28
Fifth machine (+2, ×3): 30, 36, 54, 90
Sixth machine (-1, ÷5): 3, 4, 6, 9, 12

Section B Inputs (reverse operations):

First machine (+3, ÷2): 19, 31, 43, 67
Second machine (+2, ×3): 4, 7, 12, 15
Third machine (-1, ÷4): 13, 33, 45
Fourth machine (÷5, +3): 25, 40, 60
Fifth machine (×10, -4): 4.8, 23.8, 5.4, 0.9
Sixth machine (×2, -5): 23, 11, 3.5, 0.5

Section C Missing Values:

First machine (+7, -4):
Input 2 → Output 5
Output 31 → Input 28
Input 12 → Output 15
Output 45 → Input 42

Second machine (+5, ÷3):
Input 7 → Output 4
Output 13 → Input 34
Output 12 → Input 31
Output 20 → Input 55

Third machine (-2, ×8):
Output 64 → Input 10
Output 88 → Input 13
Input 5 → Output 24
Input 11 → Output 72

Note: Some answers in Section B and C involve decimals, which are mathematically correct based on the operations given.

Final Answer: All values calculated above are correct as per the function machines described.