What is the difference between:
$(f + g)(x)$
and
$f(x) + g(x)$
?
What is the difference between:
$(f + g)(x)$
and
$f(x) + g(x)$
?
$(f+g)(x)$ is the function $f+g$ applied to the argument $x$.
$f(x)+g(x)$ is the sum of the two images of $x$ using $f$ and $g$.
That these two expressions always give the same result is the definition of $f+g$.