Question

How would you express this piece-wise function in terms of the unit-step function?

Answer 1

I got something like this:\[f\left (t\right )=\mathcal{U}\left (t\right )\left (1-t\right )+\mathcal{U}\left (t-1\right )\left (t-1\right )\]

Answer 2

Let's check it.

Answer 3

\[f(t)=\begin{cases}
-t+1 & \text{ if } 0\leq t< 1 \\
0 & \text{ if } t\geq 1
\end{cases}\]

Answer 4

\[\mathcal{L}\left \{ f(t) \right \}=\int_{0}^{1}e^{-st}(-t+1)dt+\int_{1}^{\infty}e^{-st}(0)dt\]

Answer 5

\[\mathcal{L}\left \{ f(t) \right \}=-\int_{0}^{1}e^{-st}tdt+\int_{0}^{1}e^{-st}dt+\int_{1}^{\infty}e^{-st}(0)dt\]

Answer 6

That's correct.