1. **Problem Statement:** We need to sketch the graph of the function $$y(t) = [x(t) + x(-t)]u(t)$$ where $x(t)$ is given and $u(t)$ is the unit step function. 2. **Understanding the components:** - $x(t)$ is the given piecewise function: - $x(t) = 0$ for $t < 0$ - $x(0) = 1$ - $x(t) = 2$ for $0 < t \leq 1$ - $x(t)$ decreases linearly from 2 to 0 for $1 < t \leq 2$ - $x(-t)$ is the reflection of $x(t)$ about the vertical axis. - $u(t)$ is the unit step function, which is 0 for $t < 0$ and 1 for $t \geq 0$. 3. **Calculate $x(-t)$:** - For $t > 0$, $x(-t) = x$ evaluated at negative values, so: - For $t > 0$, $-t < 0$, so $x(-t) = 0$. - For $t = 0$, $x(-0) = x(0) = 1$. 4. **Sum $x(t) + x(-t)$:** - For $t < 0$, $x(t) = 0$, $x(-t)$ is $x$ at positive $t$, which is: - For $-t > 0$, $x(-t)$ is as per the original $x(t)$: - $x(-t) = 2$ for $0 < -t \leq 1$ which means $-1 \leq t < 0$ - $x(-t)$ decreases linearly from 2 to 0 for $-2 \leq t < -1$ - For $t \geq 0$, $x(-t) = 0$ and $x(t)$ is as given. 5. **Multiply by $u(t)$:** - For $t < 0$, $u(t) = 0$, so $y(t) = 0$. - For $t \geq 0$, $y(t) = x(t) + x(-t)$, but since $x(-t) = 0$ for $t > 0$, $y(t) = x(t)$. 6. **Final function $y(t)$:** - $y(t) = 0$ for $t < 0$ - $y(0) = x(0) + x(-0) = 1 + 1 = 2$ - For $0 < t \leq 1$, $y(t) = x(t) = 2$ - For $1 < t \leq 2$, $y(t)$ decreases linearly from 2 to 0 7. **Summary:** - The graph is zero for $t < 0$ - At $t=0$, the value jumps to 2 - From $t=0$ to $t=1$, the graph is constant at 2 - From $t=1$ to $t=2$, the graph decreases linearly from 2 to 0 This matches the original $x(t)$ graph but with the value at $t=0$ doubled due to the sum. **Final answer:** $$ y(t) = \begin{cases} 0, & t < 0 \\ 2, & t = 0 \\ 2, & 0 < t \leq 1 \\ 2 - 2(t-1), & 1 < t \leq 2 \end{cases} $$