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We are tasked with finding the Laplace transform of the given piecewise function \( f(t) \), where:

\[
f(t) = 
\begin{cases}
0, & t < 2 \\
t^2 - 4t + 8, & t \geq 2
\end{cases}
\]

The Laplace transform of a function \( f(t) \) is given by:

\[
\mathcal{L}\{f(t)\} = \int_0^\infty f(t) e^{-st} \, dt
\]

### Step 1: Use the Heaviside function

We can rewrite the function \( f(t) \) using the Heaviside step function \( u(t-a) \) to manage the discontinuity at \( t = 2 \). The Heaviside function \( u(t-2) \) is 0 when \( t < 2 \) and 1 when \( t \geq 2 \). Thus, the function can be written as:

\[
f(t) = (t^2 - 4t + 8) u(t-2)
\]

### Step 2: Laplace transform of shifted function

The Laplace transform of \( f(t) = (t^2 - 4t + 8) u(t-2) \) involves shifting by 2. Using the time-shifting property:

\[
\mathcal{L}\{f(t-a) u(t-a)\} = e^{-as} \mathcal{L}\{f(t+a)\}
\]

In this case, \( a = 2 \) and \( f(t) = t^2 - 4t + 8 \), so we now find the Laplace transform of \( t^2 - 4t + 8 \), but with \( t \) shifted to \( t + 2 \).

### Step 3: Laplace transform of \( f(t) \)

Now, let's compute the Laplace transform of \( t^2 - 4t + 8 \).

We will compute this next.

The Laplace transform of the given function \( f(t) \) is:

\[
F(s) = \frac{2(2s^2 + 1)e^{-2s}}{s^3}
\]

Would you like further clarification or additional details?

Here are 5 related questions you might find interesting:
1. How do you apply the time-shifting property of the Laplace transform?
2. What is the Laplace transform of a general quadratic function?
3. Can you derive the Laplace transform of other piecewise functions?
4. How does the Heaviside function help in transforming piecewise functions?
5. What are some common applications of the Laplace transform in differential equations?

**Tip**: When dealing with piecewise functions, always try to express them using the Heaviside function to simplify the computation of their Laplace transforms.

Find the Laplace transform of the piecewise function f(t), where: f(t) = 0 for t < 2 and f(t) = t^2 - 4t + 8 for t >= 2.

Learn how to compute the Laplace transform of a piecewise function involving the Heaviside step function. This detailed solution applies the time-shifting property of the Laplace transform, with a step-by-step breakdown of the process for finding the Laplace transform of f(t) = t^2 - 4t + 8 for t >= 2.

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