Terri's project for the Math Fair was a magnificent black box that she called a function machine. If you put 3 into her machine, the output would be 8 . If you put in 10 , the output would be 29 ; and if you put in 20 , it would be 59 . Homework Help a. What would her machine do to the input 5 ? What about -1 ? What about $$ x $$ ? Making an input $$ \rightarrow $$ output table may help. b. Write an equation for Terri's machine.

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Let's analyze Terri's function machine step by step to determine its behavior.

### **Part a: Determining Outputs for Given Inputs**

We are given the following inputs and outputs:

| Input $$ x $$ | Output $$ f(x) $$ |
|--------------|-------------------|
| 3            | 8                 |
| 10           | 29                |
| 20           | 59                |

To find out what the machine does to other inputs, such as 5, -1, and a general $$ x $$, we'll first determine the underlying rule or equation of the machine.

### **Part b: Finding the Equation of the Function Machine**

Assume the function is linear, meaning it can be represented as:
$$ f(x) = ax + b $$
where $$ a $$ and $$ b $$ are constants.

Using the given inputs and outputs:

1. For $$ x = 3 $$:
   $$ 3a + b = 8 $$  
   *(Equation 1)*

2. For $$ x = 10 $$:
   $$ 10a + b = 29 $$  
   *(Equation 2)*

Subtract **Equation 1** from **Equation 2**:
$$
(10a + b) - (3a + b) = 29 - 8 \
7a = 21 \
a = 3
$$

Now, plug $$ a = 3 $$ back into **Equation 1**:
$$
3(3) + b = 8 \
9 + b = 8 \
b = -1
$$

So, the equation of the function machine is:
$$ f(x) = 3x - 1 $$

### **Calculating Specific Outputs**

Using the equation $$ f(x) = 3x - 1 $$:

1. **For $$ x = 5 $$:**
   $$
   f(5) = 3(5) - 1 = 15 - 1 = 14
   $$

2. **For $$ x = -1 $$:**
   $$
   f(-1) = 3(-1) - 1 = -3 - 1 = -4
   $$

3. **For a general input $$ x $$:**
   $$
   f(x) = 3x - 1
   $$

### **Summary**

a.  
- Input **5** produces an output of **14**.
- Input **-1** produces an output of **-4**.
- For any input $$ x $$, the output is $$ 3x - 1 $$.

b.  
The equation for Terri's machine is:
$$
f(x) = 3x - 1
$$
a. - Input **5** produces an output of **14**. - Input **-1** produces an output of **-4**. - For any input $$ x $$, the output is $$ 3x - 1 $$. b. The equation for Terri's machine is: $$ f(x) = 3x - 1 $$

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