To represent $f(t)$ in a piecewise function based on the graph provided, let's analyze the intervals and corresponding values of $f(t)$:

1. Interval Analysis:

   • For $3 \leq t < 4$: The graph has a slope, suggesting a linear relationship. Observing the points, we can deduce that the line has the form $f(t) = 2t - 2$.
   • For $4 \leq t < 8$: The graph is flat, meaning $f(t)$ is constant. From the graph, $f(t) = 6$ in this interval.
   • For values outside these intervals (i.e., $t < 3$ or $t \geq 8$): The function is 0, indicating that $f(t) = 0$ otherwise.

2. Piecewise Representation: Given the intervals and values identified, we can represent $f(t)$ as:

   2t - 2 & \text{if } 3 \leq t < 4, \\ 6 & \text{if } 4 \leq t < 8, \\ 0 & \text{otherwise}. \end{cases}$$

This representation captures the behavior of $f(t)$ as illustrated in the graph.

Would you like further details on how each part of the piecewise function was determined, or do you have additional questions?

Related Questions

1. How do you determine the slope of a linear segment on a graph?
2. Why is the function zero outside the specified intervals?
3. What are some common applications of piecewise functions?
4. How can we find the derivative of this piecewise function?
5. What does continuity mean in the context of piecewise functions?

Tip

When analyzing a graph, observe points of discontinuity or changes in slope to identify distinct intervals in piecewise functions.