https://file.aiquickdraw.com/mathbot/file/1729455440748q1q93adw.jpg

The piecewise function in the image is defined as follows:

\[
f(x) = 
\begin{cases} 
-x + 3 & \text{if } x \leq -1 \\
-3x + 1 & \text{if } x \geq -1
\end{cases}
\]

### To graph this function:
1. **For \( x \leq -1 \)**, use the equation \( f(x) = -x + 3 \), which represents a straight line with slope \(-1\) and a y-intercept at 3.
   
2. **For \( x \geq -1 \)**, use the equation \( f(x) = -3x + 1 \), which represents another straight line with slope \(-3\) and y-intercept at 1.

Both of these pieces meet at \( x = -1 \), so we should check the values of the function at this point to ensure continuity.

- For \( x = -1 \) in the first piece, \( f(-1) = -(-1) + 3 = 1 + 3 = 4 \).
- For \( x = -1 \) in the second piece, \( f(-1) = -3(-1) + 1 = 3 + 1 = 4 \).

Since the function equals 4 at \( x = -1 \) for both pieces, the function is continuous at that point.

Would you like further details or any specific clarifications?

### Here are 5 related questions to consider:
1. How do you find the slope and intercept of a linear equation?
2. What does it mean for a function to be continuous?
3. How do you determine if a piecewise function is differentiable at a point?
4. How do you solve for the domain and range of a piecewise function?
5. Can you find the intersection points of the two parts of the piecewise function?

### Tip: When graphing piecewise functions, it's important to plot key points where the function switches from one rule to another, and check for continuity or jumps at those points.

Suppose that the function f(x) is defined as a piecewise function with f(x) = -x + 3 for x <= -1 and f(x) = -3x + 1 for x >= -1.

Learn how to graph the piecewise function f(x) = -x + 3 for x <= -1 and f(x) = -3x + 1 for x >= -1. This problem covers graphing, continuity, and linear functions. Suitable for high school students.

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