Determine the Laplace Transform of the functionf(t)f(t+3)=et,0≤t<3=f(t)

The Laplace Transform of a function f(t) is defined as:

L{f(t)} = ∫_0^∞ e^(-st)f(t) dt

Given the function f(t) = e^t for 0 ≤ t < 3 and f(t+3) = f(t), we can find the Laplace Transform for each piece separately.

For f(t) = e^t, the Laplace Transform is:

L{e^t} = ∫_0^∞ e^(-st)e^t dt = ∫_0^∞ e^(t(1-s)) dt = [e^(t(1-s)) / (1-s)]_0^∞ (assuming s > 1)

This evaluates to:

= 1 / (s-1) (for s > 1)

For f(t+3) = f(t), we use the shifting property of the Laplace Transform, which states that L{e^(at)f(t)} = F(s-a), where F(s) is the Laplace Transform of f(t).

So, L{f(t+3)} = L{f(t)} = 1 / (s-1) (for s > 1)

Therefore, the Laplace Transform of the given function is 1 / (s-1) for s > 1.