F(x) as function or value of the function?

"Let's examine the function ##f(x) = x^2 + 1##" is a slightly sloppy (but much shorter) way of saying "let's examine the function ##f:\mathbb{R} \to \mathbb{R}## defined by ##f(x) = x^2 + 1##". The shorter statement doesn't generally cause any ambiguity as long as the domain and codomain are understood. It would be tedious if we always insisted on pedantic language such as "the function ##f:\mathbb{R} \to\mathbb{R}## defined by ##f(x) = e^x##" instead of simply "the function ##e^x##".

Of course, there may be cases where we need to distinguish between the function ##f## and its value ##f(x)## at some particular ##x##, and we should write more carefully in those cases.

 

##f(x)## is a notation that to implies "function ##f## is dependent of or written in terms of the variable ##x##." The notation makes it somewhat easier to see how the variable plays into determining each value of the function.

I'm going to use an example. It will involve physics because I like physics.

Let's look at the equation of the position of a massive particle moving at a constant acceleration. [tex]\vec r_f = \vec r_i + \vec v_i \Delta t + \frac{1}{2} \vec a \Delta t^2[/tex]
Since the initial position, the initial velocity, the initial time measurement, and the constant acceleration are "fixed" variables, unlike the variable ##t##, then it can be rewritten as [tex]\vec r (t) = \vec r_i + \vec v_i (t-t_i) + \frac{1}{2} \vec a (t-t_i)^2[/tex]
This shows that the particle's position ##\vec r## is dependent on time ##t##.

Authors have the ability to switch the two notations you've listed above because they can. It implies the same thing.

 

A function is a triple ##(A,B,\Gamma)## such that

1. A is a set called the domain,
2. B is a set called the co-domain,
3. ##\Gamma \subset A \times B## is a set called the graph,
4. for all ##x \in A## there exists ##(x, y) \in \Gamma## ("every function is defined on its domain"),
5. if ##(x_1, y) \in \Gamma## and ##(x_2, y) \in \Gamma## then ##x_1 = x_2## ("the vertical line test").

Notations such as ##f(x)## are attempts at defining functions without writing the whole thing down. They are just shortcuts. Different books will have different conventions.

Now as to the question concerning ##f## vs ##f(x)##. When I was much younger I came across an abstract algebra textbook focusing on ring theory dating back to the early 1930s. It used the ##xf## notation: that is ##x## is a point in the domain and ##f## is the map (homomorphism). The clear advantage is composition. ##xfg## is apply f then g. Today we have to write ##g\circ f##. This is great for algebra, horrible for applied analysis. What is ##2f##? Twice the functions? Or the function evaluated at 2? To top it off, as matrices became more standard, we have abandoned ##xf## notation for ##f(x)##, but the split between "f" and "x" stayed in pure mathematics circles. So as a consequence pure mathematicians consider the function to be "f" and the valued to be "f(x)".

But I can't find any old applied mathematics papers that doesn't use "f(x)" is the function. For an applied mathematician what a function does at each point is of prime importance. It's my belief that applied mathematics has used "f(x)" before pure mathematics has used "f" as the function.

 

Some functions sure. The identity function from ##X\to X## is the function with the property such that for any ##f:X\to X## we have ##f \circ Id = Id \circ f##.
Remember category theory can defined things like injectivity without any reference to the variable.

Majority? Depends on what information you want from it, right?

Edit for grammar

 

As you said, it usually is simplest to define a function using a dummy variable. If you think about computer programming a function is given a dummy argument so that the procedure of computing the function can be defined. After a function has been defined, it can be referred to as "the function f" or "the function f()". That notation is actually used.

The question of "what is a variable" is not simple to answer. Within a certain "scope" a variable is treated like a symbol that represents a specific number. In computer programming, the "scope" of a variable is clear. If you see an "x" within the scope of a function, you know it isn't the "x" referred to in the code for a different function. In mathematical writing, many authors are careless. For example, you may be told that f(x) = 3x + 2 and g(y) = sin(y) + y^2 are "functions". But then your are told something like y = 2x, enforcing a relation between y and x,. Many discussions in physics that speak of "functions" are actually discussion of "relations" or discussion of several different functions, all denoted as the same function. The use of specific dummy variables facilitates such ambiguous but sometimes convenient writing.