I didn't learn functional notation very well and I'm not sure exactly where to find the answer for this so if someone doesn't mind explaining some pretty simple stuff that would be greatly appreciated:
$$f(x)-g(x)=(f-g)(x)$$
What does this mean and how is it justified? Please explain as if you were explaining to a first time algebra student.
This is the definition of the difference of two functions (probably between real numbers). Suppose we have two functions $f$ and $g$. They are each given by some formula or rule that takes in a real number and outputs another real number. If the input number is $x$, we denote the output of the functions by $f(x)$ and $g(x)$ respectively.
Now we want to define a new function $f-g$ from these two functions. So we need a rule for what the output is for a certain input $x$. In words, the rule is this: plug $x$ into both $f$ and $g$. Now you have two numbers $f(x)$ and $g(x)$. Then subtract $g(x)$ from $f(x)$. This is the output for $f-g$ on $x$. As a formula this can be expressed succinctly by $(f-g)(x)=f(x)-g(x)$,
For example: if $f(1)=2$ and $g(1)=9$ then $(f-g)(1)=f(1)-g(1)=2-9=-7$
The equation $$ f(x)-g(x)=(f-g)(x) \quad (*) $$ features the functions $f$ and $g$ and a third function $f-g$ which is the difference function of $f$ and $g$.
It also uses $x$ which is some element from the domain set of $f$, $g$ and $f-g$, an argument to those functions, a value which these functions map to some element of their codomain set which is also called the image of the function:
In mathematics $f(x)$ is the value that the function $f$ maps the argument $x$ to. Example: $$ f : D \to A \\ f(x) = x^2 + 1 \in A \quad (x \in D) $$ will map an $x$ from some domain set $D$ (here unspecified) to the image set $A$. What to do with the argument $x$ to get $f(x)$ is given as an expression in this example.
The given equation $(*)$ can be read as a definition for the function $f-g$. It works by defining the function value $(f-g)(x)$ as difference of the function values $f(x)$ and $g(x)$, for arbitrary $x$ from the domain set.
Similar definitions are used to define $f + g$ (sum), $f g$ (product), $f/g$ (quotient), $f \circ g$ (composition), $c f$ (multiplication by a constant), $f'$ (taking the derivative) and more.
