# Is f(x) a function or a value?

1. Aug 25, 2015

### Nicocin

I have joined to ask a question of : is f(x) a function or a value?

2. Aug 25, 2015

### BvU

Hello Nicocin, and welcome to PF !

In case that doesn't help you much: it depends. On the context. If $f(x) = x^2$ we speak of a quadratic function. But for a specific x, for example 2.5 one can write f(x) = 2.52 = 6.25 and that is a value.

3. Aug 25, 2015

### Mentallic

Like BvU said, f(x) is a function with respect to the variable x. If x isn't a variable, but rather a specific value, say, x0 then f(x0) is a value.

4. Aug 25, 2015

### Ssnow

Hi, sometimes you use the same expression to denote the function, as the law of the variable $x$, or the value that it assumes in $x$. We can say that $f$ represent the law or function, $f(x)$ represent the value of the function in $x$ generic, $f(x_{0})$ is the value of the function in a specific point $x_{0}$ and if $f(x)=x^{4}-x^{3}$ I wrote in the right the analytic expression of the function that tell how effectively it works on the variable $x$( in this case is a polynomial expression) . If we don't know the analytic expression of the function $f$ we can say that it assume in $x_{0}=2$ the value $f(2)$. If we know the analytic expression , as before $f(x_{0})=x_{0}^{4}-x_{0}^{3}$, we can say that it assume in $x_{0}=2$ the value $f(2)=2^{4}-2^{3}$ that id the value $8$....
I hope was clear ...

5. Aug 25, 2015

### Finny

I was going to answer as did the above posters, but Wikipedia sees it differently:

In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9.

https://en.wikipedia.org/wiki/Function_(mathematics)

As post #2 says, I think it depends on the context.

6. Aug 25, 2015

### MisterX

$f$ is a function, and $f(x)$ is a value (a member of the codomain). To claim otherwise is sloppy or technically incorrect. There might be times when you want to write something like "the function $x^2$" as shorthand (having not introduced $f$), but I see no reason to write "the function $f(x)$".

7. Aug 25, 2015

### Mentallic

I do. It tells the reader that f is a function of a single variable x.

8. Aug 25, 2015

### BvU

My sincere apologies to Nicocin. It's not unusual that forum readers start haggling among themselves even over simple questions and answers, but today seems to be exceptional. Oh well, comes with the forum, all for free (and free for all) !

9. Aug 25, 2015

### axmls

$f(x)$ is the function $f$ evaluated at $x$. So it is a number, not a function.

This should usually be obvious from the context.

I think it's important to clear up the difference between the function $f$ and the number $f(x)$ precisely because the OP (like many students I've seen) does not know the difference. Functions as a concept and their notation do not make sense unless you see $f(x)$ as "the number that results when we evaluate $f$ at $x$." For students experienced in math who know the difference, saying "the function f(x)" might be okay in order to save time, but for students who are learning what functions are, it does them a disservice to be so sloppy in our wording.

You're right that it does make things a lot easier, especially in contexts where it's not clear what the independent variables should be (for instance by saying "Consider the electric potential function $V(x,y,z) = ...$"), but especially in cases where students are confused about the whole function vs number thing, it's best to be as precise as possible in our wording.

10. Aug 25, 2015

### HallsofIvy

I agree with axmls. Strictly speaking, "f" is a function, "f(x)" is a number, the specific value of f at the given value of x.

Of course, we often "abuse the notation", talking about "the function f(x)" when we really mean "the function f".

11. Aug 25, 2015

### aikismos

Hmmm. Yes, and we often define functions with mappings, so that the notation is $f : S_1 \rightarrow S_2 : S_1, S_2 ⊂ℕ$ for instance. It should also be mentioned that $f(x)$ is frequently called the range or the co-domain.

12. Aug 28, 2015

### HallsofIvy

I have never seen $f(x)$ "called the range or the co-domain". I have seen the set of all f(x) called the range or the co-domain.

13. Aug 28, 2015

### aikismos

I would suggest that the phrase 'set of all $f(x)$' is just a disambiguation of which of two things the notation is because $f(x)$ can occur in two ways:
1. $f(x) ∀x \in ℝ$ implies that $f(x)$ is a set whereas
2. $f(x) ∃!x \in ℝ$ implies that $f(x)$ is a quantity such that $(x, f(x))$
Essentially, whether $f(x)$ is one value or represents many is a matter of context which is why the existential and qualification operators appear so much in mathematical statements. I certainly concede that the use of English keeps things clear. :D

EDIT @ 10:27

I just wanted to add that this notation is typically encountered in textbooks where authors will declare $y=f(x)$. It used to puzzle the bejesus out of me in high school when I would run across the statement, but eventually what emerged was the idea that in math instruction that it's the idea of moving the thinking process into understanding there's a difference between functions and relations (because at that level, it's often not taught very thoroughly and explicitly). So, in the expression $y = f(x)$, the intent of the author's assertion is rendered in English roughly as "All values of y that come from x are now values known as f(x) because they as a set fulfill the definition of a functional range (co-domain)." I know that high school math isn't as rigorous as that of higher education, but the use of $f(x)$ as an identifier for a set is so well established, that typically the qualification 'the set of all' is not used. This, by the way, is also demonstrated by the fact that typically in notation to differentiate between $f(x) ∀x \in ℝ$ and $f(x) ∃!x \in ℝ$, one often writes $f(x) = 2x + 1$ but refers to $x_1$ and $f(x_1)$ to suggest to the reader that $(x_1, f(x_1)) \in \{ (x, f(x)) \}$. Of course, I welcome to hear any experience that differs from my own.

Last edited: Aug 28, 2015