𝑃(𝑡) =
2 + 4𝑡 𝑆𝑖 0 ≤ 𝑡 < 2
6 + 2𝑡 𝑆𝑖 2 ≤ 𝑡 < 4
14 𝑆𝑖 4 ≤ 𝑡 < 12
50 − 3𝑡 𝑆𝑖 12 ≤ 𝑡 < 16

The piecewise function \( P(t) \) is defined as follows:

\[
P(t) =
\begin{cases} 
2 + 4t & \text{if } 0 \leq t < 2, \\
6 + 2t & \text{if } 2 \leq t < 4, \\
14 & \text{if } 4 \leq t < 12, \\
50 - 3t & \text{if } 12 \leq t < 16.
\end{cases}
\]

This function changes its form based on the interval in which the variable \( t \) falls. 

Would you like to see detailed explanations of how each piece is derived or have any questions?

Here are five related questions to explore:

1. How do you evaluate \( P(t) \) for a specific value of \( t \)?
2. What is the value of \( P(t) \) at \( t = 3 \)?
3. What does the graph of \( P(t) \) look like over the interval \( 0 \leq t < 16 \)?
4. How do the slopes of the different linear segments compare across the intervals?
5. What is the physical or real-world interpretation of such a piecewise function?

**Tip:** When dealing with piecewise functions, always pay close attention to the defined intervals for \( t \) to correctly determine which expression to use for evaluation.

Explore a detailed example of a piecewise function problem and its solution. This problem involves defining a piecewise function with different linear segments, suitable for high school students studying functions and algebra.

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