Laplace Transform Piecewise Function

Question

I've never seen these types of bounds on a piecewise function of a Laplace transform before, can someone help explain how to solve this problem, particularly the Laplace transform of g(t)? Thanks in advance.

Answer 1

Hint: you can write the piecewise function using the Heaviside Unit Step function as:

$$g(t) = t - (t-3) u_3(t) = t - (t-3) u(t-3)$$

Can you now continue?

Update

To find the Laplace Transform of $g(t)$, we can rewrite it as:

$$g(t) = t - (t-3) u(t-3) = t + 3 u(t-3) - t u(t-3)$$

Using formula $28$ in the table, we have:

$$\mathscr{L} (g(t)) = \mathscr{L}(t) + 3 \mathscr{L}u(t-3) - \mathscr{L}((t+3)u(t-3)) = \dfrac{1}{s^2} + \dfrac{3}{s}e^{-3s} - \left(\dfrac{1}{s^2} + \dfrac{3}{s}\right) e^{-3s}$$