How do summations/integrals like Fourier, Laplace, z-transforms preserve all the information about the original signal?

Question

In normal summations, like 2+3=5, the information about the original numbers is lost. But in infinite summations like integral transforms, no information is lost and the function can still be recovered. What about integral transforms makes this possible? Give a less formal explanation (I mean, just avoid complex terminology. Use high school familiar terms)

Answer 1

Summations (or integrals) in discrete and continuous Fourier transforms merely takes points in one infinite dimensional space (e.g., continuous functions of a real variable) and express them as a different continuous function. No information is lost (in typical cases) because the basis set spans the space.