1. **Problem Statement:** We have four inverse function machines: - Machine 1: In → Add 8 → Out - Machine 2: In → Divide by 3 → Out - Machine 3: In → Subtract 8 → Out - Machine 4: In → Divide by 4 → Out Given outputs and inputs: Input: -2, 4, 8, 20 Output: 12 (corresponding to input 4) **Part (a):** Identify which machines produce the correct input from the given output. 2. **Step 1: Define the inverse functions for each machine.** - Machine 1: Output = Input + 8 → Inverse: Input = Output - 8 - Machine 2: Output = Input / 3 → Inverse: Input = Output × 3 - Machine 3: Output = Input - 8 → Inverse: Input = Output + 8 - Machine 4: Output = Input / 4 → Inverse: Input = Output × 4 3. **Step 2: Check which inverse machines produce the correct input for the given output 12 (corresponding to input 4).** - Machine 1 inverse: Input = 12 - 8 = 4 ✓ matches given input - Machine 2 inverse: Input = 12 × 3 = 36 ✗ does not match input 4 - Machine 3 inverse: Input = 12 + 8 = 20 ✗ does not match input 4 - Machine 4 inverse: Input = 12 × 4 = 48 ✗ does not match input 4 Only Machine 1 produces the correct input from the given output. 4. **Step 3: Complete input/output table for Machine 1 (from part a).** Using Output = Input + 8: - Input -2 → Output = -2 + 8 = 6 - Input 4 → Output = 4 + 8 = 12 - Input 8 → Output = 8 + 8 = 16 - Input 20 → Output = 20 + 8 = 28 5. **Part (b): Function machines P and Q:** - P: Output = 4 × Input - Q: Output = Input + 3 6. **Step 4: If both machines have the same input, find input such that output of P is twice output of Q.** Let input be $x$. - Output of P = $4x$ - Output of Q = $x + 3$ Condition: $4x = 2(x + 3)$ Solve: $$4x = 2x + 6$$ $$4x - 2x = 6$$ $$2x = 6$$ $$x = 3$$ 7. **Step 5: If both machines have the same output, find output if input of Q is three times input of P.** Let input of P be $a$, input of Q be $3a$. Outputs equal: $$4a = (3a) + 3$$ Solve: $$4a = 3a + 3$$ $$4a - 3a = 3$$ $$a = 3$$ Output: $$4a = 4 \times 3 = 12$$ **Final answers:** - Part (a): Only Machine 1 produces correct input from output. - Part (a) table for Machine 1: | Input | Output | |-------|--------| | -2 | 6 | | 4 | 12 | | 8 | 16 | | 20 | 28 | - Part (b) 1: Input = 3 - Part (b) 2: Output = 12