https://file.aiquickdraw.com/mathbot/file/1727455590610uibh4lzk.jpg

The image shows a piecewise function \( f(x) \) defined as:

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

### Explanation of the Function:
1. **For \( x \leq -1 \)**: The function is given by \( f(x) = x + 2 \). This is a linear function with a slope of 1 and a y-intercept of 2.
2. **For \( x > -1 \)**: The function is \( f(x) = -2x + 1 \). This is also a linear function but with a slope of -2 and a y-intercept of 1.

Would you like me to graph this function, analyze it further, or solve a specific problem related to it? 

### Additional Questions to Consider:
1. How does the graph of \( f(x) \) look at the boundary where \( x = -1 \)?
2. What is the continuity of this function at \( x = -1 \)?
3. What are the slopes and intercepts of the lines for each piece of the function?
4. How do you find the domain and range of this function?
5. What is the derivative of each piece, and what does it tell you about the function's behavior?

**Tip:** When dealing with piecewise functions, always check how the function behaves at the boundaries of each piece to ensure continuity or identify any jumps/discontinuities.

The function f(x) is defined piecewise, as shown in the image, where f(x) = x + 2 for x ≤ -1 and f(x) = -2x + 1 for x > -1.

Explore the piecewise function f(x), where f(x) = x + 2 for x ≤ -1 and f(x) = -2x + 1 for x > -1. Learn how to analyze its continuity, slopes, and y-intercepts, and understand the behavior at the boundary x = -1. Suitable for students in Grades 9-12.

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