Sketch the graph of the piecewise-defined function.
f(x) = {(x + 4 text(, ) x <= -3, -5 text(, ) -3 < x < 1, 5x - 2 text(, ) x>=1)

To sketch the graph of the piecewise-defined function:

\[
f(x) = 
\begin{cases} 
x + 4 & \text{if } x \leq -3 \\
-5 & \text{if } -3 < x < 1 \\
5x - 2 & \text{if } x \geq 1 
\end{cases}
\]

### Step-by-step breakdown:

1. **For \( x \leq -3 \):**
   - The function is \( f(x) = x + 4 \), which is a linear function with a slope of 1 and a y-intercept of 4.
   - At \( x = -3 \), the value is \( f(-3) = -3 + 4 = 1 \). This point is included since \( x \leq -3 \).
   - The line continues for \( x < -3 \).

2. **For \( -3 < x < 1 \):**
   - The function is a constant, \( f(x) = -5 \), for values between \( -3 \) and \( 1 \) (exclusive of both endpoints).
   - This means the graph will be a horizontal line at \( y = -5 \) for \( -3 < x < 1 \).

3. **For \( x \geq 1 \):**
   - The function is \( f(x) = 5x - 2 \), which is a linear function with a slope of 5 and a y-intercept of -2.
   - At \( x = 1 \), the value is \( f(1) = 5(1) - 2 = 3 \). This point is included since \( x \geq 1 \).
   - The line continues for \( x > 1 \).

### Important points:
- At \( x = -3 \), the function transitions from the first case \( x + 4 \) to the second case \( f(x) = -5 \). The point \( (-3, 1) \) is included in the first segment, but there is a discontinuity at \( x = -3 \), with no point defined in the second segment at \( x = -3 \).
- At \( x = 1 \), the function transitions from \( f(x) = -5 \) to \( f(x) = 5x - 2 \). The point \( (1, 3) \) is included in the third segment.

Would you like a detailed sketch? Let me know if you want me to visualize the graph for you or explain further!

### Additional Questions:
1. What is the importance of analyzing where the function is continuous?
2. How can you determine if a piecewise function has jump discontinuities?
3. What happens if the piecewise function has open and closed intervals at the same point?
4. How do the slopes in linear functions affect the appearance of the graph?
5. What are real-world applications of piecewise functions?

**Tip:** Always plot key points and evaluate at transition boundaries when sketching piecewise functions to visualize any discontinuities or sharp changes in the graph.

Sketch the graph of the piecewise-defined function. f(x) = {(x + 4 text(, ) x <= -3, -5 text(, ) -3 < x < 1, 5x - 2 text(, ) x>=1)

Learn how to sketch the graph of a piecewise-defined function with linear segments and discontinuities. This example covers f(x) = x + 4 for x <= -3, f(x) = -5 for -3 < x < 1, and f(x) = 5x - 2 for x >= 1. Ideal for students in Grades 9-12, with a focus on discontinuities and linear function behavior.

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