The function given is:

$f(x) = 2 \left\lfloor \frac{x}{3} \right\rfloor$

This is a step function, where $\left\lfloor \frac{x}{3} \right\rfloor$ represents the greatest integer less than or equal to $\frac{x}{3}$. Multiplying by 2 stretches the function vertically by a factor of 2. We are tasked with finding the length of each individual line segment and the positive vertical separation between each step.

Step 1: Length of Each Line Segment

Each segment of this function corresponds to a horizontal step. The function $\left\lfloor \frac{x}{3} \right\rfloor$ changes values at every multiple of 3, so the length of each line segment (along the x-axis) is 3 units.

Step 2: Vertical Separation Between Steps

The vertical jump between each step is caused by the factor of 2 in front of $\left\lfloor \frac{x}{3} \right\rfloor$. Each step increases by 2 units, so the vertical separation between each line segment is 2 units.

Final Answer:

• Length of each line segment: 3 units
• Vertical separation between steps: 2 units

Would you like more details on how this function behaves? If yes, I can walk you through a step-by-step breakdown of the graph.

Here are 5 related questions to expand on this topic:

1. How would the function change if the factor of 2 were replaced by 3?
2. What would happen if the function used $\left\lceil \frac{x}{3} \right\rceil$ (ceiling function) instead of the floor function?
3. How can you determine the range of the function based on the given step pattern?
4. What is the effect of shifting the function horizontally or vertically?
5. How do step functions relate to piecewise linear functions?

Tip: Understanding step functions is essential in many real-world applications like rounding or controlling outputs at discrete intervals.