# Function notation question

1. Jul 27, 2013

### cra18

I have seen over and over statements like:
\begin{aligned} &f(x)~\text{is a function of}\dots \\ &\text{Let}~f(x)~\text{be a function that}\dots. \end{aligned}
This is probably a dumb question, but am I justified in feeling annoyed at these statements? The annoyance stems from my understanding that the "function" is $f$, not $f(x)$, i.e., in the definition,
$$f : x \mapsto f(x),$$
so while $f$ is the literal rule that assigns a value to the point $x$, $f(x)$ is that actual value. Or am I mistaken?

2. Jul 27, 2013

### WannabeNewton

You are not mistaken. It is just an abuse of terminology.

3. Jul 27, 2013

### cra18

Thanks for your answer. But what do people mean generally? Are they referring to the rule, or the variable value of the output of the rule?

4. Jul 27, 2013

### WannabeNewton

The general meaning is that $f$ is the function, not $f(x)$; in $f:X\rightarrow Y,x \mapsto f(x)$, where $X,Y$ are sets, $f$ is the function from $X$ into $Y$ and it sends the element $x$ of $X$ to the element $f(x)$ of $Y$. People simply say things like "consider the function $f(x)$" for shorthand.