Final answer:
The functions F(s) = (s² + 4s + 8) / ((s + 1)(s + 4)²), the f(t) is -2e^(-t) + 3e^(-4t) - te^(-4t).
Step-by-step explanation:
To find f(t) given the function F(s), we need to take the inverse Laplace transform of F(s).
The Laplace transform is a mathematical tool used to transform functions of time, f(t), into functions of complex frequency, F(s).
To find f(t), we can use partial fraction decomposition to break F(s) into simpler fractions.
In this case, we have F(s) = (s² + 4s + 8) / ((s + 1)(s + 4)²).
Using partial fraction decomposition, we can write F(s) as A/(s + 1) + B/(s + 4) + C/(s + 4)².
Solving for A, B, and C, we find A = -2, B = 3, and C = -1.
Now, we can take the inverse Laplace transform of each term to find the corresponding f(t) values.
The inverse Laplace transform of A/(s + 1) is -2e^(-t), the inverse Laplace transform of B/(s + 4) is 3e^(-4t), and the inverse Laplace transform of C/(s + 4)² is -te^(-4t).
Therefore, f(t) = -2e^(-t) + 3e^(-4t) - te^(-4t).