Given Function x(t) and Period T=4 , the given function is a periodic function with period x(t) = cases2 - t & for 0 t < 2 t - 4 & for 2 t < 4cases

Let's break down the given periodic function \( x(t) \) with a period \( T = 4 \). The function seems to be defined piecewise over the interval \( t \in [0, 4) \) and repeats for every interval of length 4.

We have:

\[
x(t) = 
\begin{cases} 
2 - t & \text{for } 0 \leq t < 2 \\
t - 4 & \text{for } 2 \leq t < 4 
\end{cases}
\]

Since this function is periodic with a period \( T = 4 \), it repeats for all intervals of length 4. This means that for any \( t \), the function can be expressed as:

\[
x(t) = x(t + nT) \quad \text{for any integer } n.
\]

### Explanation of the Piecewise Function:

1. For \( 0 \leq t < 2 \): The function is \( x(t) = 2 - t \). This is a line with a slope of \(-1\) and a y-intercept of 2, which decreases from \( x(0) = 2 \) to \( x(2) = 0 \).
   
2. For \( 2 \leq t < 4 \): The function is \( x(t) = t - 4 \). This is a line with a slope of \( 1 \) and a y-intercept of \(-4\), which increases from \( x(2) = -2 \) to \( x(4) = 0 \) (but the interval ends just before 4).

Since the function is periodic with \( T = 4 \), it will repeat this pattern every 4 units of time. Thus, for any value of \( t \), we can reduce \( t \) modulo 4 to fit it within the interval \( [0, 4) \) and then apply the corresponding piecewise rule.

### Graphical Representation
If you graph \( x(t) \), it will look like a zig-zag pattern that repeats every 4 units. Specifically:
- From \( t = 0 \) to \( t = 2 \), the function decreases linearly.
- From \( t = 2 \) to \( t = 4 \), the function increases linearly.

Would you like me to go deeper into the function's properties, like Fourier series, or any other aspect of the periodic function?

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Here are five questions to deepen understanding:

1. How do you find the Fourier series representation of this periodic function?
2. What is the behavior of the function outside the interval \( [0, 4) \)?
3. How would you sketch the function for multiple periods?
4. Can you express the function in terms of a shifted Heaviside function?
5. What is the average (DC component) of this periodic function?

**Tip:** Always analyze periodic functions in terms of their fundamental period, as this simplifies calculations and visualizations across the entire time domain.

Given Function x(t) and Period T=4, the given function is a periodic function with period x(t) = { 2 - t for 0 ≤ t < 2; t - 4 for 2 ≤ t < 4 }

Explore the piecewise periodic function x(t) with period T=4, defined as x(t) = { 2 - t for 0 ≤ t < 2; t - 4 for 2 ≤ t < 4 }. Learn how to analyze its behavior, sketch its graph, and understand its properties. Suitable for advanced high school and undergraduate students.

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