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To graph the given piecewise function \( f(x) \), we need to analyze the two linear equations provided for their respective domains.

### Function Definition:
\[
f(x) = 
\begin{cases}
-2x + 4 & \text{for } x < -1 \\
-5x + 19 & \text{for } x > 3
\end{cases}
\]

### Steps to Graph:

1. **Graph \( f(x) = -2x + 4 \) for \( x < -1 \):**
   - Find a few points on this line within the domain \( x < -1 \).
   - Example points:
     - For \( x = -2 \): \( f(-2) = -2(-2) + 4 = 8 \).
     - For \( x = -3 \): \( f(-3) = -2(-3) + 4 = 10 \).
   - Plot these points and extend the line to the left of \( x = -1 \).

2. **Graph \( f(x) = -5x + 19 \) for \( x > 3 \):**
   - Find a few points on this line within the domain \( x > 3 \).
   - Example points:
     - For \( x = 4 \): \( f(4) = -5(4) + 19 = -1 \).
     - For \( x = 5 \): \( f(5) = -5(5) + 19 = -6 \).
   - Plot these points and extend the line to the right of \( x = 3 \).

3. **Verify the endpoints:**
   - Check the values of the function at the boundaries \( x = -1 \) and \( x = 3 \), if needed.
   - The function does not include these boundary points, so it should have open circles at these boundaries if you plot them.

### Graphing Tip:
To represent the function accurately, be sure to clearly show the open circles at \( x = -1 \) and \( x = 3 \) as these values are not included in the function definition.

If you need further assistance with graphing or any details, feel free to ask!

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**Relative Questions:**
1. How do you graph a piecewise function with more than two segments?
2. What are the key differences in graphing continuous vs. discontinuous piecewise functions?
3. How do you find the slope and y-intercept of each segment in a piecewise function?
4. Can a piecewise function have a constant segment, and how would you graph it?
5. How do you determine if a piecewise function is continuous at a boundary point?

**Tip:** When graphing piecewise functions, always check the inclusion (closed circle) or exclusion (open circle) of boundary points.

Graph the following function on the axes provided: f(x) = { -2x + 4 for x < -1 ; -5x + 19 for x > 3 }.

Learn how to graph the piecewise function f(x) defined as -2x + 4 for x < -1 and -5x + 19 for x > 3. This problem involves understanding piecewise functions, graphing linear equations, and using the slope-intercept form. Suitable for high school students in Grades 9-12.

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