On function names

Question

This may seem like a stupid question, but with regards to functions is $f(x)$ a function name? We write $f(x)$ for a function, where the actual algebraic formula on the right is the "machine". If we wrote something like $\sin x$ or $x+3$ without using $y=\dots$ or $f(x)=\dots$, could we say that something like $\sin x$ or $x+3$ is a function?

Answer 1

As Ethan said, $f$ is the name of the function and $f(x)$ is the number that $f$ gives you as output when you give $x$ to $f$ as input. But, people are frequently sloppy about this. In particular, you will often hear people refer to "the function $\sin(x)$" or "the function $x + 3$", for example, because it would be cumbersome to make a more correct statement such as "the function $f:\mathbb R \to \mathbb R$ defined by $f(x) = x + 3$ for all $x \in \mathbb R$".

Answer 2

Yes.

Formally speaking, a function is a relation between an input value and an output value, such that one input gives no more than one output.

There are different ways to provide function definitions, one of them being by the use of symbolic expressions, such as $\sin(x)$ or $x+3$. In these, $\sin$ denotes a "predefined" function, while $x+3$ uses an operator ($+$) which is actually a predefined function of two arguments.

You can indeed assign a name to functions, for instance by means of the notation

$$f(x):=x+3$$ or $$f(x):=\sin(x).$$

Then you can refer to the function $f$, and its evaluation at $x$, namely $f(x)$. But the RHS expressions themselves are functions (of the variable(s) they mention), even if they don't get a specific name.

$x$ denotes both the variable $x$ and the so-called "identity" function, which trivially maps $x$ to $x$. A constant such as $3$ also denotes a "constant" function, which maps all values of the variable (specified from context) to $3$.

Answer 3

The name of the function is $f$.

But there's another convention that comes into play, namely that when speaking about a function, instead of giving it a name we can just state an expression that tells how to compute the value of the function we're speaking. Thus, we can speak about "the function $x^2+3x$" for example, without giving it an explicit name.

Now, since "$f(x)$" is actually an expression that tells us to compute the value of the function named $f$, saying "$f(x)$" is also, by the above convention, a way to speak about that function. "$f(x)$" is not the name of the function, but is nevertheless a way to refer to it.

The take-home lesson from this is that in practice whether people say $f$ or $f(x)$ (when it's clear that they're speaking about a function, rather than the value of that function at some already known $x$), does not make any difference to what they mean.