The idea of a function is a rule that gives the mapping of an input number to an output number. This function is generally represented by but nowadays we like the view that the f symbolises the function (intuitively f symbolises the rule the function is applying) and is actually the number we get when we apply the function or the rule to the number . (We say is the value of f at x).

The modern idea of a function only allows rules which assign __one value__ to a particular number of x. For example if f symbolised a rule for which say and then f is not a function. Do not confuse this with a rule f which is assigning the same value to two (or more different) numbers. For example and ; this is permitted in our idea of a function.

Also when specifying a function we must not only give the rule, but also give the set of numbers to which the rule is intended to apply. This set of numbers to which the rule applies is called the domain of the function and may be denoted by or . Thus we say, define the function f by:

Now let f be a function whose domain . Then the set of values that f takes ie is called the range of f and may be denoted by or

### Some useful notation

- – the open interval a,b
- – the closed interval a,b

Also used is:

(read this as mod x where ) is defined by:

In this course the notation where will always be the positive root, as will also .

So, for example .

Note particularly the effect

In fact generally

The so called “floor function” (or integer part function) is notated as, and is defined by

eg.