https://r2.erweima.ai/mathbot/image/1726767342440l2wghqwj.jpg

This is a piecewise function \( f(x) \), defined differently for three intervals of \( x \):

\[
f(x) = \begin{cases} 
3 - 2x & \text{if } x \leq 2, \\
-1 & \text{if } 2 < x \leq 3, \\
\frac{x+1}{2} & \text{if } x > 3.
\end{cases}
\]

### Explanation of each piece:
1. **For \( x \leq 2 \):** The function is \( f(x) = 3 - 2x \), a linear expression.
2. **For \( 2 < x \leq 3 \):** The function is a constant, \( f(x) = -1 \), regardless of the exact value of \( x \) within this range.
3. **For \( x > 3 \):** The function becomes \( f(x) = \frac{x+1}{2} \), which is also a linear expression but with a different slope compared to the first piece.

Would you like a detailed analysis of specific values, continuity, or other aspects of the function?

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Here are some follow-up questions you may want to explore:
1. What is the value of \( f(x) \) at \( x = 2 \)?
2. What is the value of \( f(x) \) at \( x = 3 \)?
3. Is \( f(x) \) continuous at \( x = 2 \) and \( x = 3 \)?
4. How does the slope of \( f(x) \) change across the different intervals?
5. What is the graph of \( f(x) \) and how do the different pieces connect?

**Tip**: Always check for continuity and differentiability at the points where piecewise functions change their expressions (e.g., \( x = 2 \) and \( x = 3 \) here).

Evaluate the piecewise function f(x) defined for different intervals of x, as seen in the image.

Learn how to evaluate and analyze a piecewise function f(x) that is defined by three different expressions for distinct intervals of x. This function includes both linear and constant pieces, making it ideal for high school students learning about piecewise functions.

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