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The problem provided is asking to determine the largest interval in which the function \( y = f(2 + |x|) \) is increasing, based on the graph of \( f(x) \).

Let's break down the key steps to solve this:

1. **Understanding the function**: The given function is \( y = f(2 + |x|) \). This indicates that the function \( f(x) \) has been shifted horizontally by \( -2 \), and the input to \( f \) is modified by the absolute value of \( x \), meaning the function is symmetric about \( x = 0 \).

2. **Analyzing the graph**: The graph of \( f(x) \) shows that it is decreasing for negative \( x \), constant at a minimum value near \( x = 0 \), and increasing for positive \( x \).

3. **Key question**: The question asks in which interval the function \( y = f(2 + |x|) \) is increasing. 

4. **Effect of transformation**: The term \( 2 + |x| \) means that the behavior of \( f(x) \) is shifted to the left by 2 units and is symmetric around \( x = 0 \). The critical point where the function begins to increase will be influenced by these transformations.

### Conclusion:
By observing the graph and taking the transformations into account, the function is increasing in the interval \( [1, +\infty) \), which corresponds to the correct option.

Thus, the answer is **Option 3: \( [1, +\infty) \)**.

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Would you like further details on this solution? Here are 5 follow-up questions related to this topic:

1. How does the absolute value \( |x| \) affect the symmetry of the function?
2. What happens when you shift a function horizontally by a certain value?
3. How do you determine intervals of increase and decrease from a graph?
4. Why is the function constant in the region near \( x = 0 \)?
5. Can you explain the effect of combining transformations like absolute value and shifting?

**Tip**: Always check the behavior of the transformed function near critical points (e.g., where the absolute value or shifts take effect).

Determine the largest interval where the function y = f(2 + |x|) is increasing, based on the graph of f(x).

Analyze the graph and transformations of the function y = f(2 + |x|) to determine the largest interval of increase. This problem involves understanding graph transformations and the behavior of absolute value functions.

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