What is the difference between writing f and f(x)?

Question

I see a lot of people and professors in my calculus courses using f and f(x) in a way that looks interchangeable to me, and 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, and 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) for example, does he actually mean that x is in/dependent? Or are they just being lazy and not writing it fully? Thanks!

Answer 1

Although we are sometimes sloppy in what we write (especially when in a hurry!),

properly f describes the function, i.e. the "rule" if you like which assigns a value dependent on x to the dependent variable y = f(x)...and there you have what f(x) means; f(x) is a specific value, namely the value taken on by the dependent variable when the value of the independent variable is x.

If I write x(t), I mean that t is the independent variable and x(t) is the value which the function assigns to a dependent variable...and x in this case is the function! Usually better not to use x as a dependent variable...although especially with parametric equations it is necessary.