When is a composition $(f \circ g)(x)$ the same as $f(g(x))$

Question

Suppose $f,g$ are two functions, operators, matrices...

I noticed in some mathematical writings, people prefers $(f \circ g)(x)$ as the notation for a composition instead of $f(g(x))$ where $x$ is a vector

To me, both are compositions, just written in different forms. Is there a case when composition $(f \circ g)(x)$ and $f(g(x))$ (applying $g$ then $f$) a are not the same?

Sorry if the question sounds stupid!

Answer 1

These two notations are one and the same; the function $f\circ g$ is defined by writing $$(f\circ g)(x) :=f(g(x)).$$

The usage can be different in the sense of which part you want to emphasize.

Answer 2

The definition of $f\circ g$ is the function defined by $(f\circ g)(x) = f(g(x))$, so they are indeed the same.

Answer 3

$f\circ g$ is used to emphasize that the operation is a new function, without plugging in the value $x$. That's the significant difference. Once plugging in, they are by definition the same.