What will happen after Laplace Transform?

Question

Consider the Laplace transform $\int_{0}^{\infty} e^{-px}f(x)\,dx$
Assume $f(x)=1$ , then the Laplace transform is $\frac {1}{p}$.
Assume $f(x)=x$ , then the Laplace transform is $\frac {1}{p^2}$.
The question is, what will happen to the $f(x)$ after getting transformed?
Why should the function be transformed and what aspect of initial function will remain in the Laplace transform that makes it so important?
If someone can give geometric intuition of it, it will be a plus!
Thanks

Answer 1

One major advantage lies in the fact that many differential equations become algebraic equations when the Laplace transform is applied. We can then solve the algebraic equations and take the inverse Laplace transform (the transform is one to one and so has an inverse) to arrive at a solution to the differential equation.

To see how the transform relates to infinite series and other connections see http://en.wikipedia.org/wiki/Laplace_transform#Relation_to_power_series

Answer 2

The Laplace transform ${\cal L}$ is applied solely to known or unknown functions which are defined explicitly or implicitly in terms of "analytic formulas". What makes ${\cal L}$ useful are alone its formal algebraic properties, encoded in certain rules of term manipulation. A central ingredient of the Laplace philosophy is Lerch's theorem which says that ${\cal L}$ is injective. So, when you have found a solution $s\mapsto Y(s)$ in transform space it suffices to look up the unique function $t\mapsto y(t)$ whose transform is $Y$ in a suitable catalogue.

Don't hope for an "intuitive content" encoded in ${\cal L}f$. Nobody has ever looked at the graph of an ${\cal L}f$, or has computed ${\cal L}f$ for an $f$ which is only defined by a data set. This is in sharp contrast to the Fourier transform: Of course we work with it all the time in theoretical discussions, but apart from that the Fourier transform $\hat f$ of a time signal $f$ conveys interesting "intuitive information" about $f$, and people are Fourier-transforming discreetly sampled time signals all the time.