Is $f\circ g(x) = (f\circ g)(x)$? Do they have similar properties on how they work as a functions.
The function $(f\circ g)(x)$ is equal to $f(g(x))$. But I am unaware on how $f\circ g(x)$ works.
My guess is that $f\circ g(x)$ is the same as $(f\circ g)(x)$ since you compose the $x$ in $(f\circ g)(x)$ together to achieve $f \circ g(x)$. But a problem that i noticed that if we compose $(f\circ g)(x)$ it together as I stated to get $f\circ g(x)$, it would also imply that you can achieve $f(x)\circ g$
Due to that problem, I am unaware what is the appropriate answer. Furthermore, I am unable to find anything online.
First note that $f\circ g (x) $ means $f\circ (g (x)) $ and this is, generally, different from $(f\circ g)(x) $. Indeed, $f\circ g (x) $ is a function when $g (x) $ is a function. More precisely, let $g:X\to Z^Y $ and $f:Z\to W $. Then for all $x\in X$ we have $g (x):Y\to Z $ hence $f\circ g (x):Y\to W $ and $$(f\circ g (x))(y)=f (g (x)(y)) $$