f(x)=y is called function notation while `f:RR->RR;x|->y` (often shortened to `f:x|->y` ) is called mapping notation.
A function can be considered a "rule" that assigns an argument (called the preimage) from the domain to an argument (called the image) in the codomain. In the example `f:RR->RR;x|->y` we say f is a function mapping from the reals to the reals where f maps x to y (x is associated with y.)
Often the domain/codomain is left out (usually when understood in context) and you have `f:x|->y` .
The visual imagery associated with mapping diagrams can be helpful when describing functions, e.g. determining if the function is one-to-one, onto, or both. It can also be useful when describing the inverse relation for a given function (is the inverse a function also?) and when describing composition of functions.
Note that there are functions without explicit rules -- these functions can be described with tables or sets which imply using a mapping diagram.
`f(x) = x^2` is the same as `f:x rArr x^2` (by way of example)
The `(x)` in the `f(x)` is what is called the "argument." Often a value will be approtioned; for example,
`f(2)=x^2` becomes `f(2)= (2)^2 = 4` as the `x=2`
Ans: f(x) `=x^2` can also be written as f:x `rArrx^2`