Find the Laplace transform f(s) for the figure showing below

Question

Figure

I tried to solve it and came up with this solution.

Answer 1

The unit step function can have many interpretations depending on what sources your looking at.

$$f(t)=\begin{cases} 2, & \text{if $t$ t: [0,1]} \\ 1, & \text{if $t$ t: [1,2]} \end{cases}$$

for $t\in [0,1]$, $F(s)=\int_0^\infty 2e^{-st} dt = \frac{s}2$. After taking the inverse Laplace $\implies$ $f(t)=2$

for $t\in [1,2]$ $F(s_1)=\int_0^\infty 1e^{-s_1t_1} dt = \frac{s_1}1$ where $t_1=t-1$. After inverse Laplace $\implies$ $f=f(t_1)=1$

Since $t_1=t-1$ you can substitute $t(t_1)=t_1+1$ into f which becomes $f=1$.