To sketch the function $ f(x) = \frac{|x| + 2}{x^2 + 1} $, we will summarize the key points and characteristics derived from previous steps: 1. **Domain**: The function is defined for all $ x \in (-\infty, \infty) $. 2. **Y-intercept**: The y-intercept is at $ (0, 2) $. 3. **X-intercepts**: There are no x-intercepts since $ f(x) $ cannot equal zero. 4. **Stationary Points**: - Local maximum at $ x = -2 - \sqrt{5} $. - Local minimum at $ x = -2 + \sqrt{5} $. 5. **Increasing and Decreasing Intervals**: - Increasing on $ (-\infty, -2 - \sqrt{5}) $ and $ (-2 + \sqrt{5}, \infty) $. - Decreasing on $ (-2 - \sqrt{5}, -2 + \sqrt{5}) $. 6. **Concavity**: - The second derivative test will indicate intervals of concavity and convexity. Without explicit calculation, we can assume typical behavior based on the nature of rational functions and the presence of absolute values. ### Sketching Steps: 1. **Plot the y-intercept**: Mark the point $ (0, 2) $ on the graph. 2. **Mark the stationary points**: - Calculate $ f(-2 - \sqrt{5}) $ and $ f(-2 + \sqrt{5}) $ to find their corresponding y-values. - For $ x = -2 - \sqrt{5} $ (approximately $ -4.236 $), calculate: $$ f(-2 - \sqrt{5}) = \frac{|-2 - \sqrt{5}| + 2}{(-2 - \sqrt{5})^2 + 1} $$ - For $ x = -2 + \sqrt{5} $ (approximately $ -0.764 $), calculate: $$ f(-2 + \sqrt{5}) = \frac{|-2 + \sqrt{5}| + 2}{(-2 + \sqrt{5})^2 + 1} $$ 3. **Draw the function**: - Start from the left, increasing towards the local maximum at $ x = -2 - \sqrt{5} $. - Decrease until reaching the local minimum at $ x = -2 + \sqrt{5} $. - Then, increase again towards the right. 4. **Label important points**: - Mark the local maximum and minimum points on the graph. - Indicate the increasing and decreasing intervals with arrows. 5. **Concavity**: - If the second derivative indicates concavity, adjust the curvature of the graph accordingly. ### Example Sketch: Here is a rough sketch representation: ``` | 3| * | / \ 2|--------*---*-------- (0, 2) | / \ 1| / \ | / \ | / \ 0|---*-------------*---------------- | -5 -4 -3 -2 -1 0 1 2 | ``` - The asterisks (*) represent the local maximum and minimum points. - The curve increases to the maximum, decreases to the minimum, and then increases again. ### Conclusion: This sketch captures the essential features of the function $ f(x) $. The exact shape may vary based on the precise calculations of the stationary points, but the overall behavior is consistent with the analysis provided. With the above info i need a detailed and a neat graphical representation