Final answer:
To determine the Laplace transform of the function f(t)=[t-1 + e^-(t-1)]u(t-1), the time-shifting theorem is applied after finding the Laplace transforms of t and e^-t, which are then adjusted for the unit time shift.
Step-by-step explanation:
To determine L[f(t)] for the given function f(t) = [t-1 + e-(t-1)] u(t-1), where u(t-1) is the unit step function, we can apply the time-shifting theorem. The Laplace transform of a function f(t) shifted in time by a units, where the shift is to the right, is given by L[f(t-a)u(t-a)] = e-asL[f(t)], where s is the complex frequency variable.
For the given function, the time shift is 1 unit to the right, so a = 1. The function without the shift is f(t) = t + e-t. Therefore, we want to find the Laplace transform of t and e-t and then apply the shift.
The Laplace transform of t is L[t] = 1/s2, and the Laplace transform of e-t is L[e-t] = 1/(s+1). Applying the shift, we have:
- L[tu(t-1)] = e-s/s2
- L[e-(t-1)u(t-1)] = e-s/(s+1)
Therefore, L[f(t)] = e-s(1/s2 + 1/(s+1)). This is the Laplace transform of the given function after applying the time-shifting theorem.