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

To sketch the graph of the piecewise function: \[ g(x) = \begin{cases} x + 1, & \text{if } x \leq -4 \\ \frac{1}{2}x - 5, & \text{if } x > -4 \end{cases} \] let's analyze each part of the function and understand how they connect. ### 1. Analyzing Each Piece #### a. For \( x \leq -4 \): \( g(x) = x + 1 \) - **Type of Function**: This is a linear function with a slope of 1 and a y-intercept at (0, 1). - **Key Point at \( x = -4 \)**: \[ g(-4) = (-4) + 1 = -3 \] - **End Behavior**: As \( x \) decreases (goes to the left), \( g(x) \) decreases linearly. #### b. For \( x > -4 \): \( g(x) = \frac{1}{2}x - 5 \) - **Type of Function**: This is also a linear function but with a slope of \( \frac{1}{2} \) and a y-intercept at (0, -5). - **Limit as \( x \) Approaches -4 from the Right**: \[ \lim_{x \to -4^+} g(x) = \frac{1}{2}(-4) - 5 = -2 - 5 = -7 \] - **End Behavior**: As \( x \) increases (goes to the right), \( g(x) \) increases gradually. ### 2. Continuity at \( x = -4 \) At \( x = -4 \), we need to check if the function is continuous. - **Left-hand Limit** (\( x \to -4^- \)): \[ \lim_{x \to -4^-} g(x) = (-4) + 1 = -3 \] - **Right-hand Limit** (\( x \to -4^+ \)): \[ \lim_{x \to -4^+} g(x) = -7 \] - **Function Value**: \[ g(-4) = -3 \] Since the left-hand limit (\(-3\)) does not equal the right-hand limit (\(-7\)), there is a **jump discontinuity** at \( x = -4 \). ### 3. Sketching the Graph Below is a descriptive sketch of the graph: ``` y | | * | / | / | / | / | / | / | / | / | / | / | / | / | / |-----------*-------------- x | (-4, -3) (Jump to (-4, -7)) | ``` **Explanation**: - **For \( x \leq -4 \)**: Plot the line \( y = x + 1 \). It passes through the point \((-4, -3)\). This line extends to the left. - **For \( x > -4 \)**: Plot the line \( y = \frac{1}{2}x - 5 \). The value approaches \(-7\) as \( x \) approaches \(-4 \) from the right, but at \( x = -4 \), the function value is \(-3\). This creates a jump from \(-3\) to values approaching \(-7\). - **Discontinuity**: At \( x = -4 \), there's an open circle at \((-4, -7)\) and a closed circle at \((-4, -3)\) to indicate the jump. ### 4. Key Points to Plot - **For \( x \leq -4 \)**: - Point at \( (-4, -3) \) - Another point at \( (-5, -4) \) (since \( g(-5) = -5 + 1 = -4 \)) - **For \( x > -4 \)**: - Point approaching \( (-4, -7) \) as \( x \) approaches \(-4\) from the right - Point at \( (0, -5) \) (y-intercept) - Point at \( (2, -4) \) (since \( g(2) = \frac{1}{2}(2) - 5 = -4 \)) ### 5. Summary - **Two distinct linear segments**: - A steeper line for \( x \leq -4 \) - A less steep line for \( x > -4 \) - **Jump Discontinuity at \( x = -4 \)** - **No overlapping points**; the function switches from one line to the other abruptly. This piecewise function models a situation where the behavior changes at \( x = -4 \), resulting in two separate linear trends connected by a jump.