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.

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: \[ f(x) = 3x - 1 \]