What is the difference between writing f and f(x)?

[SHASHWAT PRATAP SING]

Some people say f is the function and some say f(x) is the function. f and f(x) are often used interchangeably. I have seen in many textbooks, sentences such as "Let f(x) be a function ...". But as we know that f represents our function then why it is given "f(x) is a function". i am confused between these two.
In many common texts, we define functions as f(x), such as f(x)=2x.

But what i know is f denotes our function.So, if f is the name given to our function, and our function is 2x. then can't we write f=2x meaning our function is 2x and its name is f.
Please help me i am confused between f and f(x)...

 

[fresh_42]

Your question is about notation. Notations vary among areas and authors. Let us assume calculus as the area, in which case the author can be neglected as they basically all use the same notation.

$f$ is the name of the function, e.g. $f\, : \,\mathbb{R}\longrightarrow \mathbb{R}$.
This function $f$ may map $x$ to $y$, i.e. $f\, : \, x\longmapsto y$ which is the same as $f(x)=y$.
We do not distinguish between $f$ and $f(x)$ in calculus in the sense that both are called the function $f$. This is a bit sloppy, because $f(x)$ only represents a single value in the range, namely the image of $x$ under the function $f$, but it has the great advantage that it simultaneously says $x$ is the general name of the variable, where we can plug in the input values of the domain.

If you really want to understand what a function is, then you have to learn what a relation is. A function is a special relation.

It makes not much sense to bother too much about whether $f$ or $f(x)$ is the function. It is only a notation. It is important what it does, what it distinguishes from a relation, what makes it well defined, and what the role of the variable is. $f$ or $f(x)$ is not important.

 

[etotheipi]

Maybe you'll find this helpful, from Proof by Hammack,

Suppose $A$ and $B$ are sets. A function $f$ from $A$ to $B$ (denoted as $f: A \rightarrow B$) is a relation $f \subseteq A \times B$ from $A$ to $B$ satisfying the property that for each $a \in A$ the relation $f$ contains exactly one ordered pair of form $(a,b)$. The statement $(a,b) \in f$ is abbreviated $f(a) = b$.

 

[SHASHWAT PRATAP SING]

f is the name of the function, e.g. f:R⟶R.
This function f may map x to y, i.e. f:x⟼y which is the same as f(x)=y.
We do not distinguish between f and f(x) in calculus in the sense that both are called the function f. This is a bit sloppy, because f(x) only represents a single value in the range, namely the image of x under the function f, but it has the great advantage that it simultaneously says x is the general name of the variable, where we can plug in the input values of the domain.

Thanks for help i understand that f denotes the function and f(x) denotes the value of f when plugging in x for the argument of the function. Speaking of "the function f(x)" is what mathematicians call "abuse of notation.
But, yes The notation f(x) to denote a function remains because it is often more convenient, and it is especially prominent from high school mathematics up to calculus because it is psychologically easier to become accustomed to.

fresh_42 if i want to represent the function only by name. For example can i write f=2x meaning my function is 2x and i have named it f.

 

[fresh_42]

@fresh_42 if i want to represent the function only by name. For example can i write f=2x meaning my function is 2x and i have named it f.

No, $f=2x$ would be wrong, because it is confusing. You can either write $f(x)=2x$ or $f\, : \,x\longmapsto 2x$. If you only write $f=$ then you didn't name a variable. $2x$ is thus only an arbitrary value and $f=2x$ could as well mean $f(t)\equiv 2x$, i.e. $f$ is a constant function which maps everything to the value $2x$. $x$ is not automatically the name of a variable. To make it one, we need the notation $f(x)$ which is why it is used. $f$ alone is just an element of a subset of a cartesian product of two sets, and nothing else can be concluded from it. Only the word function carries information in that case.

 

[SHASHWAT PRATAP SING]

No, $f=2x$ would be wrong, because it is confusing. You can either write $f(x)=2x$ or $f\, : \,x\longmapsto 2x$. If you only write $f=$ then you didn't name a variable. $2x$ is thus only an arbitrary value and $f=2x$ could as well mean $f(t)\equiv 2x$, i.e. $f$ is a constant function which maps everything to the value $2x$. $x$ is not automatically the name of a variable. To make it one, we need the notation $f(x)$ which is why it is used. $f$ alone is just an element of a subset of a cartesian product of two sets, and nothing else can be concluded from it. Only the word function carries information in that case.

Thanks for your help.Really it has helped me a lot.

fresh_42 sorry if this is a silly question but what does this notaion mean f:x⟼2x. can you please explain this notation f:x⟼2x .It would be very kind of you.

 

[fresh_42]

Thanks for your help.Really it has helped me a lot.

fresh_42 sorry if this is a silly question but what does this notaion mean f:x⟼2x. can you please explain this notation f:x⟼2x .It would be very kind of you.

There are three steps when we define a function:

1. Choose a name for the function. Say we want to define the function $g$.
2. Next we need to name domain and codomain, from where where to. Say $g\, : \,[0,1]\longrightarrow \mathbb{R}$ where the interval $[0,1]$ is the domain, and the real numbers $\mathbb{R}$ the codomain of our function $g$. Range is usually the subset of the codomain which is the image under $g$, i.e. $$\operatorname{im} g =\operatorname{range}(g)=\{r\in \mathbb{R}\,|\,\text{ there exists a } t\in [0,1]\text{ such that }g(t)=r\}.$$
   I'm not completely certain whether all authors distinguish between range and codomain, so maybe the range is occasionally the entire codomain, regardless whether a number is hit by $g$ or not.
3. Now that we have the name of the function ($g$), where it is defined on $([0,1])$, and where it is aimed at ($\mathbb{R}$), we finally need to know what it exactly does on a given input. Since we do not want to and cannot write all possible inputs $g(0)=... ,g(0.1)=..., g(0.2)=...$ etc, for all possible inputs $t\in [0,1]$ we first need a variable name, say $t$. So $t$ is any number form the intervall $[0,1].$ But what is the output? Where does $g$ map such a $t$ to? This can be written as, say $g(t)=4t+2$, or as $g:t\longmapsto 4t+2$ which reads: $g$ maps $t$ onto $4t+2.$

Summary: Let $g$ be a function defined as
\begin{align*}
g\, : \,[0,1]&\longrightarrow \mathbb{R}\\
t&\longmapsto 4t+2
\end{align*}
$g\, : \,t \longmapsto 4t+2$ is just another way to say this, if the domain and codomain is clear, e.g. if we talk about function from $\mathbb{R}$ to $\mathbb{R}$ all the time.

 

[SHASHWAT PRATAP SING]

There are three steps when we define a function:

1. Choose a name for the function. Say we want to define the function $g$.
2. Next we need to name domain and codomain, from where where to. Say $g\, : \,[0,1]\longrightarrow \mathbb{R}$ where the interval $[0,1]$ is the domain, and the real numbers $\mathbb{R}$ the codomain of our function $g$. Range is usually the subset of the codomain which is the image under $g$, i.e. $$\operatorname{im} g =\operatorname{range}(g)=\{r\in \mathbb{R}\,|\,\text{ there exists a } t\in [0,1]\text{ such that }g(t)=r\}.$$
   I'm not completely certain whether all authors distinguish between range and codomain, so maybe the range is occasionally the entire codomain, regardless whether a number is hit by $g$ or not.
3. Now that we have the name of the function ($g$), where it is defined on $([0,1])$, and where it is aimed at ($\mathbb{R}$), we finally need to know what it exactly does on a given input. Since we do not want to and cannot write all possible inputs $g(0)=... ,g(0.1)=..., g(0.2)=...$ etc, for all possible inputs $t\in [0,1]$ we first need a variable name, say $t$. So $t$ is any number form the intervall $[0,1].$ But what is the output? Where does $g$ map such a $t$ to? This can be written as, say $g(t)=4t+2$, or as $g:t\longmapsto 4t+2$ which reads: $g$ maps $t$ onto $4t+2.$

Summary: Let $g$ be a function defined as
\begin{align*}
g\, : \,[0,1]&\longrightarrow \mathbb{R}\\
t&\longmapsto 4t+2
\end{align*}
$g\, : \,t \longmapsto 4t+2$ is just another way to say this, if the domain and codomain is clear, e.g. if we talk about function from $\mathbb{R}$ to $\mathbb{R}$ all the time.

Thanks fresh_42 for helpng me.

 

[PeroK]

Some people say f is the function and some say f(x) is the function. f and f(x) are often used interchangeably. I have seen in many textbooks, sentences such as "Let f(x) be a function ...". But as we know that f represents our function then why it is given "f(x) is a function". i am confused between these two.
In many common texts, we define functions as f(x), such as f(x)=2x.

But what i know is f denotes our function.So, if f is the name given to our function, and our function is 2x. then can't we write f=2x meaning our function is 2x and its name is f.
Please help me i am confused between f and f(x)...

Just to add one point. It would be good if there were a way to refer to specific functions without using the function value. But, you can only do this to some extent. You could say: let $f$ be the sine function or $g$ be the exponential function. But, if you want to have: $$f(x) = \sin (2x + 1)$$ then there is no practical way to talk about that function except using its function value.

 

[robphy]

Possibly interesting reading:
https://bridge.math.oregonstate.edu/papers/FEdgap.pdf
https://bridge.math.oregonstate.edu/papers/bridge.pdf

 

[Delta2]

The way I know it , f is the function and f(x)=... is the formula of the function f. To fully define a function we need to define the domain, the codomain and the formula. However, some authors define the function just by defining the formula f(x), because the domain and codomain are easily implied or can be found from the context.