Graph the following function on the axes provided \[ f(x)=\left\{\begin{array}{lll}x-1 & \text { for } & 2 \leq x<6 \\ -3 & \text { for } & x=6 \\ -4 & \text { for } & x>6\end{array}\right. \]

Certainly! Let's graph the piecewise function: \[ f(x)=\left\{ \begin{array}{lll} x-1 & \text{for} & 2 \leq x < 6 \\ -3 & \text{for} & x = 6 \\ -4 & \text{for} & x > 6 \end{array} \right. \] ### Step-by-Step Graphing: 1. **Identify the Different Pieces of the Function:** - **Line Segment:** \( f(x) = x - 1 \) for \( 2 \leq x < 6 \) - **Point:** \( f(x) = -3 \) at \( x = 6 \) - **Horizontal Line:** \( f(x) = -4 \) for \( x > 6 \) 2. **Plot Each Piece Separately:** - **For \( 2 \leq x < 6 \), \( f(x) = x - 1 \):** - Calculate endpoints: - At \( x = 2 \): \( f(2) = 2 - 1 = 1 \) - As \( x \) approaches 6: \( f(6^-) = 6 - 1 = 5 \) - Plot a **solid line** from \( (2, 1) \) to just below \( (6, 5) \). Use an **open circle** at \( (6, 5) \) to indicate that \( x = 6 \) is not included in this interval. - **At \( x = 6 \), \( f(x) = -3 \):** - Plot a **solid dot** at \( (6, -3) \). - **For \( x > 6 \), \( f(x) = -4 \):** - Draw a **horizontal dashed line** starting just after \( x = 6 \) at \( y = -4 \). - Optionally, use an **open circle** at \( (6, -4) \) if you want to indicate that \( x = 6 \) is not included in this interval. 3. **Combine All Pieces:** Here's a conceptual representation of the graph: ``` y | 5 | o 4 |----------------- 3 | * 2 | 1 |● 0 |________________________ x 2 6 ``` - **●** at \( (2, 1) \): Start of the line segment. - **o** at \( (6, 5) \): Open circle where the line segment approaches but doesn't include \( x = 6 \). - **\*** at \( (6, -3) \): Solid dot representing \( f(6) = -3 \). - **\--** line extending to the right from \( x = 6 \) at \( y = -4 \): Represents \( f(x) = -4 \) for \( x > 6 \). ### Detailed Description: - **From \( x = 2 \) to \( x = 6 \):** - The function increases linearly with a slope of 1. - It starts at \( (2, 1) \) and approaches \( (6, 5) \) but does not include this point. - **At \( x = 6 \):** - There is a distinct value \( f(6) = -3 \), represented by a closed dot. - **For \( x > 6 \):** - The function is constant at \( y = -4 \). ### Summary: - **Line Segment:** Solid line from \( (2,1) \) to \( (6,5) \) (open at \( x = 6 \)). - **Point at \( x = 6 \):** Closed dot at \( (6,-3) \). - **Horizontal Line for \( x > 6 \):** Dashed or solid line at \( y = -4 \). If you have graphing software or graph paper, you can plot these points and lines accordingly to visualize the function.