Convolution Notation: The difference between (f*g)(x) and f(x)*g(x)

Question

What is the difference between (f*g)(x) and f(x)*g(x) [1] for convolutions? Are they the same? I ask this because I have been asked to prove the Reflection of Convolution property for my course in the Theory of Distributions, i.e that f(-x)g(-x)=(fg)(-x). But if there is no difference between [1], then surely the proof follows from [1] and nothing actually needs to be proven.

Answer 1

$f*g$ means that $*$ is defined on the space of functions. The notation $f(x)*g(x)$ makes it look as though $*$ is defined on the space of real numbers (if the functions are real that is), and hence isn't great. It's the same solecism as talking about 'the function $f(x)$'---it's $f$ that is the function, not $f(x)$.