https://file.aiquickdraw.com/mathbot/file/1727877768552bwypp6ap.jpg

Let's analyze the piecewise function \( f(x) \):

\[
f(x) =
  \begin{cases} 
   x - 1 & \text{for } x < -3, \\
   -5 & \text{for } -3 \leq x \leq 1, \\
   4x & \text{for } x > 1.
  \end{cases}
\]

### 1. First interval: \( x - 1 \) for \( x < -3 \)
This is a line with a slope of 1 and a y-intercept of -1. For \( x < -3 \), this line will be graphed only to the left of \( x = -3 \).

### 2. Second interval: \( -5 \) for \( -3 \leq x \leq 1 \)
This is a constant function, which means it will be a horizontal line at \( y = -5 \) for \( x \) between -3 and 1, inclusive of both endpoints.

### 3. Third interval: \( 4x \) for \( x > 1 \)
This is another linear function with a slope of 4 and a y-intercept of 0. It will only be graphed to the right of \( x = 1 \).

### Now, let's match the options:

- **Option A**: This graph appears to show a correct piecewise function where:
  - \( x - 1 \) for \( x < -3 \) is sloping downward (incorrect slope).
  - The constant function \( y = -5 \) between \( -3 \leq x \leq 1 \) is shown correctly.
  - \( 4x \) for \( x > 1 \) appears correct.

- **Option B**: Seems to match the correct behavior:
  - The line \( x - 1 \) for \( x < -3 \) has the correct slope.
  - The constant function \( y = -5 \) is in place for \( -3 \leq x \leq 1 \).
  - The line \( 4x \) starts correctly from \( x = 1 \).

- **Options C and D**: Both of these contain inaccuracies in how the piecewise function is presented, such as incorrect slopes or misplacements of the constant segment.

Thus, **Option B** is the correct graph for this piecewise function.

Would you like more details on how each interval was graphed? Also, here are some related questions for further understanding:

1. How do you graph a piecewise function with multiple intervals?
2. What is the significance of open and closed dots in piecewise graphs?
3. How do you determine the slope of a linear function?
4. Why is there a break in the graph of a piecewise function?
5. How does the constant function behave in piecewise functions?

**Tip**: Always check the domain restrictions for each part of a piecewise function to ensure you graph only within the specified intervals.

Graph the piecewise function f(x) = { x-1 for x < -3, -5 for -3 ≤ x ≤ 1, 4x for x > 1 }, and choose the correct graph.

Learn how to graph the piecewise function f(x) with three intervals: f(x) = x - 1 for x < -3, f(x) = -5 for -3 ≤ x ≤ 1, and f(x) = 4x for x > 1. This step-by-step solution includes analyzing the behavior of each function segment and determining the correct graph representation. Ideal for Grades 9-12.

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