[Math] Laplace transform of piecewise continuous function

Question

$$f(t) =\begin{cases}t^2 & 0 \le t < 3,\\ 9& t \ge 3\end{cases}$$

1. Show that $f$ is of exponential order.
2. Express $f$ in terms of the unit step function.
3. Find Laplace transform of $f$ and determine the allowed values for
   $s$.

I am not sure how to write piecewise function so I cannot begin to solve the problem.

Answer 1

Hint: You can write this using Heaviside Unit Step functions (plot this versus your piecewise function) as:

$$f(t) = t^2(u_0(t) - u_{3}(t)) + 9(u_{3}(t))$$

The Laplace Transform of this is:

$$\mathscr{L} (f(t)) = \dfrac{2}{s^3}-\dfrac{e^{-3 s} (9 s^2+6 s+2)}{s^3}+\dfrac{9 e^{-3 s}}{s}$$