I was having a look at this post: What is $f(t) * g(-t)$ (convolution)? and had my initial question answered. However, another one shows up.
I know the definition
$$ (f*g)(x) = \int_{-\infty} ^{+\infty} f(\tau)g(x-\tau)d\tau $$
and according to the post I linked above, I know that
$$ f(x)*g(-x) = \int_{-\infty} ^{+\infty} f(\tau)g(\tau-x)d\tau, $$
but is there any shorter way to write this last formula? As before it is written $(f*g)(x)$ instead of $f(x)*g(x)$.
Another question is: what lies behind such an equality? I know that the convolution between two functions is used to measure the similarity between them. But what is the point computing things like $f(x)*g(-x)$ or any other convolution where the variable in brackets isn't the same on both sides?
I apologize if my question has already been asked before. I've looked for it but without success.