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

This is probably a stupid question, and apologies if it is xD

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 :)

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That's analysts for you. They're scared of using proper notation. –  Git Gud Jan 13 '13 at 0:51
–  Hans Lundmark Jan 13 '13 at 13:33

It's not a stupid question. It's actually quite valid. Due to heavy abuse of notation (that is often harmless, though confusing), $f$ and $f(x)$ are often used interchangeably. Formally, $f:A \to B$ is a certain kind of subset of the cartesian product $A \times B$. A little less formally, $f$ is a rule that assigns to each $a \in A$ a unique value $b \in B$. We often denote this unique value as $f(a)$. So $f(a)$ is the function $f$ evaluated at some point $a$, while $f$ is actually the more abstract object that associates elements of $A$ to elements of $B$.

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Oh, I see...so I guess f would be (incorrectly) used as f(x), even if f(x) is used again in the same equation but is actually written as f(x) in that part of it? E.g. f + y = f(x) / t ? –  lunrSabr Jan 13 '13 at 0:57
Reminds me of JavaScript, where functions are variables. –  zzzzBov Jan 13 '13 at 2:45
@zzzzBov: This is no less true in mathematics. Given the right structure, you can add, subtract, and multiply functions, or calculate the distance or angle between two functions. –  Dietrich Epp Jan 13 '13 at 6:34
In Calculus especially, this ambiguity can be a source of confusion. For example, we usually think of $f(x/2)$ as the abstract object you get when you compose division by 2 with $f$, since there is no convenient alternative notation for the former. But then $f'(x/2)$ usually means the derivative of $f$ evaluated at $x/2$, not the derivative of $f \circ \frac{\cdot}{2}$. –  user7530 Jan 13 '13 at 6:51

Sometimes people write something like $f(x)$ for a function and $f(s)$ for its Laplace transform, and then the question is, does $f(3)$ mean the original function evaluated at $x=3$ or the Laplace transform evaluated at $s=3$? The point is that $f(x)$ should refer to the value of the function when the argument (or "input") to the function is the number called $x$.

Similarly, some write $f(x)$ and $f(y)$ for the probability density functions of two random variables called (capital) $X$ and (capital) $Y$. So what's $f(3)$? The point again, is one shouldn't do that; $f(x)$ should refer to the value of the function when the argument (or "input") to the function is the number called $x$. A better notation is $f_X(x)$ where (capital) $X$ is the random variable and (lower-case) $x$ is the argument to the function. Then it's clear what $f_X(3)$ is and what $f_Y(3)$ is.

Then if you write about $f(x)$ and $f(w)$, you've got the same function evaluated at two different arguments. What is the same is $f$.

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