Function notation question

[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?

 

[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.

 

[CellCoree]

I understand your frustration with these statements as they may not align with your understanding of function notation. However, it is important to recognize that in mathematics, function notation is often used interchangeably and both forms, f and f(x), are correct and valid representations of a function.

In some cases, using f(x) may be more convenient as it explicitly shows the input variable x. Additionally, it can also be used to represent the output value of the function at a specific input, f(x). On the other hand, using just f may be preferred when discussing the function as a whole, without specifying a specific input or output.

Ultimately, the choice to use f or f(x) is a matter of preference and both forms are accepted in mathematics. It is important to understand and be comfortable with both notations in order to fully grasp the concept of functions.