I am confused about Laplace transform. the only thing that is important in Laplace theorem is the value of $f(t)$ in $t \geq 0$.
laplace transform:
$\mathcal{L}\{f(t)\}$ $=$ $\int_0^\infty \! e^{-st}f(t)$
so as we see in the above transform $f(t)$ if $t<0$ is not important at all. so do we find the answer for only $t > 0$ or the answer is correct in $\mathbb{R}$?
In your setting, yes you will find your transform for $t \geq 0$. If you're interested in $\mathbb{R}$ you do $$ F(s) = \int_{-\infty}^\infty e^{-st} f(t)$$ which is also known as Bilateral (2-sided) Laplace transform.