Let's break down and solve the problem step by step: ### 1. **Step Function Definition** \[ y = \begin{cases} 3 & \text{if } -5 < x \leq -1 \\ 1 & \text{if } -1 < x \leq 3 \\ -1 & \text{if } 3 < x \leq 7 \\ -3 & \text{if } 7 < x \leq 11 \end{cases} \] ### 2. **Graphing the Step Function** - From \( x = -5 \) to \( x = -1 \) (not including -5, including -1): \( y = 3 \) - From \( x = -1 \) to \( x = 3 \) (not including -1, including 3): \( y = 1 \) - From \( x = 3 \) to \( x = 7 \) (not including 3, including 7): \( y = -1 \) - From \( x = 7 \) to \( x = 11 \) (not including 7, including 11): \( y = -3 \) Here is how the graph would look: ``` y ^ | 3 |----●========● | -5 -1 | 1 | ○=======● | -1 3 | -1| ○======● | 3 7 -3| ○=======● | 7 11 +--------------------------------> x ``` - Solid circles (●) indicate the endpoint **included**. - Open circles (○) indicate the endpoint **not included**. - Each segment is horizontal and jumps to the next at the endpoints. ### 3. **Domain and Range** - **Domain:** The union of all intervals where the function is defined: \[ (-5, -1] \cup (-1, 3] \cup (3, 7] \cup (7, 11] \] - **Range:** The set of possible output values: \[ \{3, 1, -1, -3\} \] --- If you want a digital graph, let me know!