Given the piecewise function definition of $x(t)$, let's break it down step-by-step to clarify it:

1. For $t \geq 0$: The function $x(t) = 0.5 \cdot t \cdot u(t)$. Here, $u(t)$ represents the unit step function. The unit step function $u(t)$ is defined as:

   \begin{cases} 1, & \text{if } t \geq 0 \\ 0, & \text{if } t < 0 \end{cases}$$ Therefore, for $$t \geq 0$$, we have: $$x(t) = 0.5 \cdot t \cdot 1 = 0.5 \cdot t$$

2. For $t < 0$: Since $u(t) = 0$ for $t < 0$, this implies:

   $x(t) = 0.5 \cdot t \cdot 0 = 0$

Thus, the piecewise function for $x(t)$ is:

\begin{cases} 0.5 \cdot t, & \text{if } t \geq 0 \\ 0, & \text{if } t < 0 \end{cases}$$ ### Finding $$x(0)$$: At $$t = 0$$: $$x(0) = 0.5 \cdot 0 = 0$$ Thus, $$x(0) = 0$$. ### Summary: The function $$x(t)$$ is defined as: $$x(t) = \begin{cases} 0.5 \cdot t, & \text{if } t \geq 0 \\ 0, & \text{if } t < 0 \end{cases}$$ And $$x(0) = 0$$. Would you like further details or explanations? ### Related questions: 1. How does the unit step function influence $$x(t)$$ in this case? 2. What is the graphical representation of $$x(t)$$ for this function? 3. How can $$x(t)$$ be used in signal processing? 4. What happens to $$x(t)$$ if we modify the scaling factor from 0.5 to another constant? 5. How would the function change if the unit step function $$u(t)$$ had a delay, like $$u(t - T)$$? **Tip**: The unit step function is commonly used to define signals that "turn on" at specific times, especially in control systems and signal processing.