Sketch a graph of \( f(x)=\left\{\begin{array}{ll}-3 & \text { if } x \leq-1 \\ x & \text { if }-12\end{array}\right. \)

Certainly! Let's break down how to sketch the graph of the piecewise function: \[ f(x) = \begin{cases} -3 & \text{if } x \leq -1 \\ x & \text{if } -1 < x \leq 2 \\ 1 & \text{if } x > 2 \end{cases} \] ### Step-by-Step Guide to Sketching the Graph 1. **Identify the Different Regions:** - **Region 1:** \( x \leq -1 \) where \( f(x) = -3 \). - **Region 2:** \( -1 < x \leq 2 \) where \( f(x) = x \). - **Region 3:** \( x > 2 \) where \( f(x) = 1 \). 2. **Plot Each Region Separately:** - **Region 1 (\( x \leq -1 \)):** - This is a horizontal line at \( y = -3 \). - **Endpoint at \( x = -1 \):** - Since \( x = -1 \) is included in this region, place a **closed circle** at \( (-1, -3) \). - **Region 2 (\( -1 < x \leq 2 \)):** - This is the line \( y = x \), which is a straight line with a slope of 1 passing through the origin. - **Start Point just greater than \( x = -1 \):** - At \( x = -1 \), \( f(x) \) approaches \( -1 \) from the right, but \( x = -1 \) is not included in this region. Place an **open circle** at \( (-1, -1) \) to indicate that \( x = -1 \) is not part of this segment. - **Endpoint at \( x = 2 \):** - \( x = 2 \) is included, so place a **closed circle** at \( (2, 2) \). - **Region 3 (\( x > 2 \)):** - This is a horizontal line at \( y = 1 \). - **Start Point just greater than \( x = 2 \):** - At \( x = 2 \), the previous region reaches \( y = 2 \), but \( x = 2 \) is not included in this region. Place an **open circle** at \( (2, 1) \). 3. **Combine All Regions:** - **From \( x = -\infty \) to \( x = -1 \):** Draw a horizontal line at \( y = -3 \) with a closed circle at \( (-1, -3) \). - **From \( x = -1 \) to \( x = 2 \):** Draw the line \( y = x \) starting just above \( x = -1 \) (with an open circle at \( (-1, -1) \)) and ending at \( (2, 2) \) with a closed circle. - **From \( x = 2 \) to \( \infty \):** Draw a horizontal line at \( y = 1 \) starting just after \( x = 2 \) (with an open circle at \( (2, 1) \)). ### Visual Representation While I can't provide a visual graph here, here's a textual representation to help you visualize: ``` y | 3 | 2 | ●-------- 1 | ○--------- 0 | ○ -1|●------------------------- |____________________________ x -1 0 2 ``` - **●** represents a closed circle. - **○** represents an open circle. - The horizontal lines are at \( y = -3 \) and \( y = 1 \). - The diagonal line represents \( y = x \) between \( x = -1 \) and \( x = 2 \). ### Key Points to Remember - **Closed Circle (●):** The point is included in the graph. - **Open Circle (○):** The point is not included in the graph. - **Continuity:** Check for any jumps or gaps in the graph at the boundary points (\( x = -1 \) and \( x = 2 \)). By following these steps, you should be able to accurately sketch the graph of the given piecewise function.