https://file.aiquickdraw.com/mathbot/file/1729472297341wic63s1l.jpg

The uploaded image shows a piecewise-defined function \( f(x) \) given by:

\[
f(x) =
\begin{cases}
x + 2 & \text{if } x \leq -1 \\
-x^2 + 2x & \text{if } -1 < x < 1 \\
\frac{1}{x} & \text{if } x > 1
\end{cases}
\]

Would you like to analyze the properties of this function? I can help with the following:

1. **Domain and range**  
2. **Continuity at points** like \( x = -1 \) and \( x = 1 \)  
3. **First derivative** for checking differentiability or finding critical points  
4. **Graph sketch**  
5. **Limits** as \( x \to -1 \), \( x \to 1 \), and \( x \to \infty \)

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Do you have any specific analysis in mind or shall we start with continuity and domain? Here are five questions that can expand on this:

1. What are the points where the function might not be continuous?  
2. Is the function differentiable at \( x = -1 \) and \( x = 1 \)?  
3. What are the limits of \( f(x) \) as \( x \) approaches the boundaries?  
4. How do we find the critical points of the quadratic part?  
5. What is the behavior of \( f(x) = \frac{1}{x} \) as \( x \to \infty \)?

**Tip:** Piecewise functions are often tested for **continuity** and **differentiability** at points where the rules change.

What is the behavior of the piecewise function f(x), defined as f(x) = x + 2 for x ≤ -1, f(x) = -x^2 + 2x for -1 < x < 1, and f(x) = 1/x for x > 1?

Explore the behavior of a piecewise function defined by linear, quadratic, and reciprocal parts. Analyze continuity at key points, differentiability, and critical points of the function. Suitable for advanced high school or early college students.

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