Write
$f(t) = \begin{cases} 5,& \mbox{if} \quad 0 \leq t \lt 3 \\ -4,& \mbox{if} \quad 3 \leq t \lt 7 \\ 0,& \mbox{if} \quad t \geq 7 \end{cases}$
as a unit step function and find the Laplace transform.
Workings:
The unit step function is
$f(t) = 5 + u(t-3)(-9) + u(t-7)(4)$
$f(t) = 5 - 9u(t-3) + 4u(t-7)$
The Laplace transform would then be:
$\mathcal L \{f(t)\} = \mathcal L \{5\} - 9 \mathcal L \{u(t-3)\} + 4 \mathcal L \{u(t-4)\}$
$\mathcal L \{f(t)\} = \frac{5}{s} - \frac{9e^{-3s}}{s} - \frac{e^{-4s}}{s}$
I'm not sure if this is correct. Any help will be appreciated.
As you have, $f(t)=5u(t-0)-9u(t-3)+4u(t-7)$, so
$\displaystyle\mathcal L\{f(t)\}=5\cdot\frac{1}{s}-9\cdot\frac{e^{-3s}}{s}+4\cdot\frac{e^{-7s}}{s}=\frac{5}{s}-\frac{9}{s}e^{-3s}+\frac{4}{s}e^{-7s}.$