(Solved): 3. In this problem we aim to find the Laplace transform of function f(t) given in Figure 1 through ...

3. In this problem we aim to find the Laplace transform of function $f(t)$ given in Figure 1 through the following steps: (a) Let $g(t)$ be the first triangle over $t?[0,2]$ in Figure 1, i.e., $g(t)=?\begin{array}{ll}t& \text{ if }t?[0,1]\\ 2?t& \text{ if }t?[1,2]\\ 0& \text{ else. }\end{array}$ Find $G(s)=\mathcal{L}[g(t)]$. (b) Write $f(t)$ over $t?[0,?)$ as an infinite sum of $g(t)$ and its time delays, and then use linearity of Laplace transform and the time-delay property to compute $F(s)=\mathcal{L}[f(t)]$.