In functions and notation what is the difference between f(x) and f:x ?     Totally muddled - please help!



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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.

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