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So while studying i encountered a laplace transform for a piecewise function. Now the instructions are to solve this using heavyside without the use of integrals.

$$f(t) = \begin{cases} 3t,& \mbox{if} \quad 0 \lt t \leq 1 \\ 3,& \mbox{if} \quad 1 \lt t \leq 3 \\ 12-3t,& \mbox{if} \quad 3 \lt t \leq 4 \\ 0,& \mbox{if elsewhere} \end{cases}$$

frankly i have no idea how to start

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This is only a structured hint.

I'll denote by $U(t-a)$ the Heaviside function which is $1$ when $t \geq a$, zero otherwise. Now if we define the function $g(t)$ to assume values of $f(t)$ on the interval $[a, b]$, then we may write

$$ g(t) = f(t) (U(t-a) - U(t-b)) $$

and so the Laplace transform of $g$ is

$$ G(s) = e^{-as}F(s-a) - e^{-bs}F(s-b) $$

(prove it with the integral definition, it's a good exercise).

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