FUNCTIONS : Theoretical doubt on functions 2 In the functional mathematics language , if i represent function by $$f$$ . 
What is the theoretical difference between$$f$$ and $$f(x)$$ ?
Please provide a lucid explanation.Thanks.
 A: A function is a machine, a machine that assigns to any value of some set $X$ a unique element $f(x)$ beloging to some set $Y$. So the proper notation would be $f:X\rightarrow Y:x\mapsto f(x)$. You read it as follows: "$f$ is a function from $X$ to $Y$ which assigns to any $x\in X$ a value $f(x)$ in $Y$". When you write $f$ you refer to the function, when you write $f(x)$ you refer to the element $f(x)\in Y$. Sometimes, by abuse of notation, we write $f(x)$ to refer to the function $f$ and stress that $f$ is dependend on one variable $x$.
A: When we use $f$ it just shows a function . But when we use $f(x)$ it means function at point x.
Let $f:\mathbb{N} \rightarrow \mathbb{R}$ defined by $f(x)=x+1$ .
In this example we have function $f$ and at point $x \in \mathbb{N}$ it has value $f(x)=x+1$
A: Strictly speaking, any symbol can be anything. You should however use symbols properly. Let me explain.
There are situations in which $f(x)$ is really a function. Consider, just to give an example, the differential of a function $f$ at a point $x$: it is the map $Df(x) \colon h \mapsto Df(x)h$. Essentially we have a function attached to each point $x$. In this case there is no abuse of notation.
On the contrary, it is an abuse of notation to refer to a function $f$ as $f(x)$. It is an abuse because you are confusing very different objects: a function and a point (or number). Furthermore, this abuse is dangerous because so many students cannot deal with functions defined by a different letter than $x$. I have had students who were unable to differentiate the function defined by $f(t)=\sin t$ because there was no $x$!
