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!
Best Answer
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 $B$.