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The given problem asks to graph the function \( f(x) \) defined piecewise as:

\[
f(x) = \begin{cases} 
\frac{1}{8}x, & \text{for } x < 0 \\
x + 3, & \text{for } x \geq 0
\end{cases}
\]

Let's analyze each piece:

### 1. \( f(x) = \frac{1}{8}x \) for \( x < 0 \):
- This is a linear function with a slope of \( \frac{1}{8} \). 
- Since \( x < 0 \), this part will be graphed on the left side of the y-axis.
- As \( x \) approaches 0 from the left, \( f(x) \) will approach 0.

### 2. \( f(x) = x + 3 \) for \( x \geq 0 \):
- This is another linear function, but with a slope of 1 and a y-intercept of 3.
- For \( x = 0 \), \( f(0) = 3 \), so the graph starts at the point \( (0, 3) \).
- As \( x \) increases, the function increases linearly.

### Points to consider for graph matching:
- The graph should have two distinct parts: a line with a shallow slope (for \( x < 0 \)) and a steeper line starting at \( (0, 3) \) for \( x \geq 0 \).
- The graph must be continuous at \( x = 0 \), meaning both lines should meet there. At \( x = 0 \), \( f(x) \) jumps from the value of \( \frac{1}{8}(0) = 0 \) to \( 0 + 3 = 3 \), which is consistent with the behavior in the piecewise function.

Looking at the options, **Option B** is the correct graph. It displays:
- A line with a small positive slope for \( x < 0 \).
- A line with a slope of 1 starting from \( (0, 3) \) for \( x \geq 0 \).

Would you like further explanation or additional details?

Here are five follow-up questions:
1. What is the slope of the line for \( x \geq 0 \)?
2. How would the graph change if the slope for \( x < 0 \) were steeper?
3. Why does the graph have a discontinuity at \( x = 0 \)?
4. How can we check that both parts of the function meet at \( x = 0 \)?
5. What would happen to the graph if the function for \( x \geq 0 \) was \( 2x + 3 \) instead?

**Tip:** In piecewise functions, always check the behavior at the boundary points to ensure the graph matches the function's definition at those critical points.

Graph the piecewise function f(x) = {1/8x, for x < 0; x + 3, for x ≥ 0}. Choose the correct graph from the options A, B, C, D.

Solve a piecewise function involving f(x) = 1/8x for x < 0 and f(x) = x + 3 for x ≥ 0. Learn how to match the correct graph based on the function's linear segments. Suitable for students in Grades 8-10.

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