Here is the graph of the function \( f(x) = \frac{|x| + 2}{x^2 + 1} \).

The plot includes:

- **Y-intercept** at \( (0, 2) \), marked with a red dot.
- **Local maximum** at \( x = -2 - \sqrt{5} \approx -4.236 \), marked with a green dot and labeled.
- **Local minimum** at \( x = -2 + \sqrt{5} \approx -0.764 \), marked with a green dot and labeled.

The graph shows the general shape of the function, illustrating its key features:

1. The function increases towards the local maximum at \( x = -4.236 \).
2. Decreases until reaching the local minimum at \( x = -0.764 \).
3. Increases again as \( x \) moves to the right.

Would you like more details or have any questions?

### Questions:
1. What is the significance of the local maximum and minimum points in this function?
2. How does the behavior of \( f(x) \) change near the y-intercept?
3. Why does \( f(x) \) have no x-intercepts?
4. Can we derive the exact intervals of concavity and convexity for this function?
5. How does the absolute value in the numerator affect the shape of the graph?
6. What are the implications of the function being defined for all \( x \)?
7. How can we verify the increasing and decreasing intervals for this function?
8. What role does the denominator play in the behavior of \( f(x) \)?

### Tip:
When analyzing functions, graphing them provides valuable insights into their behavior, including intercepts, maxima, minima, and points of inflection.

Explore the graph of the function f(x) = (|x| + 2) / (x^2 + 1), highlighting key features such as y-intercept, local maxima, and local minima. Understand the impact of absolute value on the function's shape and its behavior across different intervals. Suitable for advanced high school students studying functions and graph analysis.

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