Visualizing the Fourier and Laplace transform

Question

I know that $$F(\omega):=\mathscr{F}\{f(t)\}(\omega)=\int\limits_{t=-\infty}^{\infty}f(t)e^{i\omega t}\mathrm{d}t$$ $$L(s):=\mathscr{L}\{f(t)\}(s)=\int\limits_{t=0}^{\infty}f(t)e^{-st}\mathrm{d}t$$ But what's the 'visual' connection between the function $f(t)$ and it's Laplace transform $L(t)$ and $f(t)$ and it's Fourier transform $F(t)$? Like in the case of Fourier series; it represents the function as a linear combination of sinusoids.