Sketch the graph of the function. \[ g(x)=\left\{\begin{array}{ll}x+1, & x \leq-4 \\ \frac{1}{2} x-5, & x>-4\end{array}\right. \] \[ \text {, } \]

To sketch the graph of the piecewise function: \[ g(x) = \begin{cases} x + 1, & \text{if } x \leq -4 \\ \frac{1}{2}x - 5, & \text{if } x > -4 \end{cases} \] follow these steps: ### 1. Analyze Each Piece Separately **a. For \( x \leq -4 \):** - **Function:** \( g(x) = x + 1 \) - **Type:** Straight line with a slope of **1** and a y-intercept at **(0, 1)**. - **Key Point at \( x = -4 \):** \[ g(-4) = -4 + 1 = -3 \] So, the point **(-4, -3)** is included in this segment. **b. For \( x > -4 \):** - **Function:** \( g(x) = \frac{1}{2}x - 5 \) - **Type:** Straight line with a slope of **½** and a y-intercept at **(0, -5)**. - **Behavior Near \( x = -4 \):** \[ \lim_{x \to -4^+} g(x) = \frac{1}{2}(-4) - 5 = -2 - 5 = -7 \] This means as \( x \) approaches **-4** from the right, \( g(x) \) approaches **-7**. However, the point **(-4, -7)** is **not** included in this segment since \( g(x) \) is defined as \( x > -4 \) here. ### 2. Determine Continuity at \( x = -4 \) - **Left Limit:** \( g(-4) = -3 \) - **Right Limit:** \( \lim_{x \to -4^+} g(x) = -7 \) Since the left and right limits at \( x = -4 \) are not equal, there is a **jump discontinuity** at this point. ### 3. Plotting the Graph **a. Plotting \( g(x) = x + 1 \) for \( x \leq -4 \):** - Draw a straight line with slope **1**. - Include the point **(-4, -3)** with a **solid dot** (since \( x = -4 \) is included). **b. Plotting \( g(x) = \frac{1}{2}x - 5 \) for \( x > -4 \):** - Draw a straight line with slope **½**. - Exclude the point **(-4, -7)** by placing an **open circle** at this point. **c. Additional Points for Clarity:** - For \( g(x) = \frac{1}{2}x - 5 \), when \( x = 0 \), \( g(0) = -5 \). Plot the point **(0, -5)**. - For \( g(x) = x + 1 \), when \( x = -5 \), \( g(-5) = -4 \). Plot the point **(-5, -4)**. ### 4. Graph Overview - **Left Segment (\( x \leq -4 \)):** A line rising to the right with a slope of **1**, passing through **(-4, -3)**. - **Right Segment (\( x > -4 \)):** A gentler line rising to the right with a slope of **½**, starting just above **x = -4** with an open circle at **(-4, -7)** and passing through **(0, -5)**. - **Discontinuity:** A visible jump from **(-4, -3)** to just above **(-4, -7)**. ### 5. Visual Representation While I can't provide a visual sketch here, you can visualize the graph based on the description: - **Left Line:** Starts from the left, increases with a slope of 1, and ends at **(-4, -3)** with a solid dot. - **Right Line:** Begins just to the right of **x = -4**, approaching **(-4, -7)** with an open circle, then continues with a slope of ½ through points like **(0, -5)**. Remember to clearly indicate the solid and open dots to show where each piece of the function is defined.