What does "$f$ is a function of $x$" mean?

Question

Rigorously speaking, a function $f: X \to Y$ is a mapping that maps each element from set $X$ to an element in set $Y$. (Or more rigorously it can be defined using cartesian product). For $x \in X$, people often say $f$ is a function of $x$. What does that mean？Isn't $x$ just an element in $X$ and functions (in my understanding) are meant to be dealing with the entire set $X$.

Answer 1

The complete statement would be "$f$ is a function of $x\in X$". This literally means that $$f\colon X \to ?, $$ where $?$ has not been specified. Often, it is obvious from context that $x\in X$, hence the abbreviated version "$f$ is a function of $x$".

Answer 2

In the expression "$f$ is a function of $x$" a lot is left unsaid.

What that usually means is that "$x$" is the name for a "typical" element of the domain, which may or may not be explicitly specified and may or may not be called "$X$".

In some context you might say "pressure $P$ is a function of temperature $T$" when dealing with the physics of the gas law $$ PV = nRT . $$