Transform vs function?

Question

Is there any difference between transform and function? Forexample Laplace transform can be considered a function that transform a differential equation to algebraic equation

Answer 1

It's imprecise, and not really logical, but: in the traditional terminology a function maps a number to a number, but a transform maps a function to a function. So $f\in L^2(\mathbb R)$, say, is a function (sort of), because to each $x\in\mathbb R$ there is (sort of) a numerical value $f(x)$. But the map that sends $f\in L^2(\mathbb R)$ to $\hat f\in L^2(\mathbb R)$ by $\hat f (y) = \int_{\mathbb R} \exp(-2\pi i xy)f(x)dx$, say, is a transform. In other words, a transform is a map from a function space to another function space. Next up: a functional is a map from a function space to numbers.

Answer 2

I think that many people use the two terms more or less interchangeably, but one distinction I've heard is that transforms are functions where the domain and codomain are the same set. So transforms are a subset of functions.