Using the step and shift theorems to find the Laplace transform

Question

The Problem

$$f(t):=\begin{cases}e^{-t}&t\lt4\\e^{-2t}&4\le t\le10\\0&t\gt10\end{cases}$$

What I know I can write this as one function:

$$f(t)= e^{-t} + (e^{-2t} - e^{-t})u(t-4) + (0 - e^{-2t})u(t-10)$$

and the Laplace property of:

$$g(t)u(t-a) = e^{-as}L(g(t+a))(s) $$

Question Which property can I use for this transformation?

Answer 1

Unless you are asked to use some properties, you can work out the problem from scratch as

$$ \int_{0}^{4} \dots e^{-st}\, dt + \int_{4}^{10} \dots e^{-st}\, dt + \int_{10}^{\infty}\dots e^{-st}\, dt .$$

Now, plug in the right integrands in the above