TITLE: What is the difference between writing $f$ and $f(x)$?
QUESTION [26 upvotes]: I see a lot of professors in my calculus courses using $f$ and $f(x)$ in a way that looks interchangeable. Sometimes it drives me crazy because I always thought of them as being different. ($f$ means an independent variable, $f(x)$ means a variable which is dependent on $x$.) I also can't keep up with which variable is dependent on which...
So, when a professor writes down $f$ instead of $f(x)$ or $x$ instead of $x(t)$, do they actually mean that $x$ is in/dependent? Or are they intentionally not writing it fully?
Thanks!

REPLY [5 votes]: Usually $f$ and $f(x)$ are used interchangeably. But rigorously they should be used in the following way.
The single letter $f$ denotes the function, i.e. the machine in which you can plug in a value and get another value out. Yes, this might be confusing at first, because you are accustomed to using single letters for variables. However, variables can be of all kind of types. There are numbers, letters, truth values, ..., and functions. Functions are just another possible type of variables. The fact that $f$ is a function must be stated somewhere in the text before it is used. E.g. like this:

Let $f:A\to B$ be a function. Then $f$ is ...

After this line you just have to remember that it is a function. In contrast, $f(x)$ denotes the value of $f$ when plugging in $x$ for the argument of the function. Usually, $f(x)$ is not a function, but whatever $f$ spits out (e.g. a real number).
Example. Let $f:\Bbb R\to\Bbb R$ be a function defined by $f(x)=x^2$. 
Then $f$ denotes the "process of squaring", while $f(x)$ is just another way to write $x^2$, and $f(2)$ is just another way to write $4$. Also the variable used to be plugged into $f$ has no fixed name. So $f(s)$ means $s^2$ and no different function than $f$. However, you can see all kind of abuse of this notation, e.g. denoting the Laplace transform of $f$ by $f(s)$.