Function Notation

Definitions:

• A function is a way of mapping each input to exactly one output.
• The value we put into a function is called the input.
• The value that comes out is known as the output.
• A mapping is a rule that relates the inputs and outputs.

Here are the four types of mappings:

          One-to-one: One input corresponds to                      One-to-many: One input corresponds to
                                one output                                                                    many outputs

     Many-to-one: Many inputs correspond to                 Many-to-many: Many inputs correspond to
                                one output                                                                    many outputs

To note, something that gives out more than one output is not a function. Therefore, one-to-many and many-to-many mappings are not functions. 

There are two ways in which we describe functions:

1)  where   is the output and is the input.
2)  

The colon ‘:’ means ‘maps to’. Therefore, this statement reads  ' maps each input to .'

Here are some set notations that are often used alongside functions:

• - means 'is an element of'

• - The natural numbers {, , , , …}

• - The integers {…, , , , , , …}

• - The positive integers, which are equivalent to the natural numbers.

• - The rational numbers - These are all numbers that can be written as a fraction with integer numerators and denominators.

• -  The real numbers - These are all possible numbers on the number line including irrational numbers such as and .

• - The complex numbers (we’ll save this for further maths!)