https://file.aiquickdraw.com/mathbot/file/1727806875089ooktrxk1.jpg

We are given the piecewise function:

\[
f(x) = 
\begin{cases} 
2x, & \text{for } x < 0 \\
x + 5, & \text{for } x \geq 0 
\end{cases}
\]

We need to identify the correct graph for this function. Let's analyze the behavior of the function on each interval:

### 1. For \( x < 0 \):
The function is \( f(x) = 2x \), which is a straight line with slope 2 passing through the origin. It applies only for values of \( x \) that are negative (left side of the y-axis).

### 2. For \( x \geq 0 \):
The function is \( f(x) = x + 5 \), which is another straight line with a slope of 1 and a y-intercept of 5. This applies for values of \( x \) that are greater than or equal to 0 (right side of the y-axis), and includes the point \( (0, 5) \).

### Key Features to Check:
- For \( x < 0 \), the line should follow \( 2x \), which means a steep line passing through the origin but only on the left of the y-axis.
- For \( x \geq 0 \), the line should start at \( (0, 5) \) and have a slope of 1 (a gentler slope).

### Now, let’s examine the options:
- **Option A**: This shows a line starting at \( (0, 5) \) on the right but does not match the slope on the left side.
- **Option B**: This graph has the correct lines for both the left and right sections.
- **Option C**: Incorrect for the left side since the slope does not match.
- **Option D**: The slope on the right side is incorrect.

### Conclusion:
The correct graph is **Option B**, as it correctly reflects the behavior of both parts of the piecewise function.

Would you like further explanation or details on any part of the function? 

Here are five follow-up questions:
1. How do you find the slope of a line given its equation?
2. What is the significance of the y-intercept in a linear function?
3. How do you determine where a piecewise function changes its rule?
4. What happens to the graph of a function when you change the slope?
5. How does continuity apply to piecewise functions?

**Tip**: When graphing piecewise functions, always check the slope and y-intercept for each segment, and note where the rule changes.

Graph the piecewise function: f(x) = {2x, for x < 0; x + 5, for x >= 0}. Choose the correct graph.

Analyze and graph a piecewise function where f(x) = 2x for x < 0 and f(x) = x + 5 for x >= 0. This problem highlights concepts in algebra such as linear equations, slopes, and y-intercepts. Suitable for Grades 9-11.

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