What does it mean to add functions $f$ and $g$ without $x$? I can verify all the axioms, but I don't understand them. For example, the axiom of commutativity: 
$$f(x)+g(x)=(f+g)(x)=(g+f)(x)=g(x)+f(x).$$   
If the above is correct, what does it mean to add $f$ and $g$ without $x$? (e.g. add exponential function to square root function)
P.S. I'm teaching myself linear algebra and the book does not have a solution manual, so my question might be ridiculously easy! Apologize in advance.
 A: $f+g$ is just a symbol to denote the resulting function given by $f(x)+g(x)$.
e.g.
$f(x)=\sin x$
$g(x)=x$
$(f+g)(x)=\sin x+x$
A: Note that $f$ is a function, whereas $f(x)$ is the value of the function $f$ at the point $x$ in the domain of $f$. 
The point is that while we are quite comfortable with the notion of adding real numbers to obtain a new one, we have to decide what we mean by adding functions. First of all, the sum of two functions should again be a function (just like the sum of two real numbers is a real number). So what is the function obtained by adding $g$ to $f$, which we denote by $f+g$? Well, a function is determined by its values, so we need to specify $(f+g)(x)$. Now we define 
$$(f+g)(x) := f(x) + g(x).$$
Note that $f(x)$ and $g(x)$ are real numbers, so we know what it means to add them. As before, $f+g$ is the name of the function, and $(f + g)(x)$ is the value of the function $f + g$ at the point $x$ in its domain. But what is its domain? Well, $f$ and $g$ are both defined on $[0, 1]$, so we take the domain of $f + g$ to be $[0, 1]$ as well. More generally, we take the domain of $f + g$ to be $\operatorname{Dom}(f)\cap\operatorname{Dom}(g)$.
