determine a function f(t) that has the given Laplace transform F(s). F(s) = frac{3}{(s^2)}

determine a function f(t) that has the given Laplace transform F(s).
$F\left(s\right)=\frac{3}{\left({s}^{2}\right)}$
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Step 1
Given,
$F\left(s\right)=\frac{3}{\left({s}^{2}\right)}$
Step 2
Formula:
$\left(1\right){t}^{n}={L}^{-1}\left\{\frac{\left(n!\right)}{{s}^{n+1}}\right\}$
Constant Multiplication property:
$\left(2\right){L}^{-1}\left\{aF\left(s\right)\right\}=a\cdot {L}^{-1}\left\{F\left(s\right)\right\}$
Step 3
Consider, $F\left(s\right)=\frac{3}{\left({s}^{2}\right)}$
Apply Laplace inverse transform on both sides
${L}^{-}1\left\{F\left(s\right)\right\}={L}^{-1}\left\{\frac{3}{\left({s}^{2}\right)}\right\}$
$f\left(t\right)={L}^{-1}\left\{3\cdot \frac{1}{\left({s}^{2}\right)}\right\}$

$f\left(t\right)=3\cdot {L}^{-1}\left\{\frac{1!}{{s}^{1+1}}\right\}$

$f\left(t\right)=3t$
Therefore, $f\left(t\right)=3t$ that gives a Laplace transform $F\left(s\right)=\frac{3}{\left({s}^{2}\right)}$