Is It Correct to Use f and f(x) Interchangeably in Mathematics?

[cra18]

I have seen over and over statements like:
<br /> \begin{aligned}<br /> &amp;f(x)~\text{is a function of}\dots \\<br /> &amp;\text{Let}~f(x)~\text{be a function that}\dots.<br /> \end{aligned}<br />
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,
<br /> f : x \mapsto f(x),<br />
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?

 

[WannabeNewton]

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

 

[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?

 

[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.