# Functions And Graphing Functions

Basics:

A function is a rule that changes input into output

A relation is any set of ordered pairs

A function is defined as a set of ordered pairs in which no two ordered pairs have the same element

A function must give exactly one unique output for each input

Also called a mapping or simply a map

The set of input numbers is called the domain

The set of output numbers is called the range

The set of all possible outputs is called the co-domain

The range is generally the subset of the co-domain however they can also be the same

Brackets:

A domain described as

That is, the square bracket means p is included. The rounded bracket means q is not included.

Number systems:

Composite functions:

When one function is followed by another function, the result is a composite function

Applying function after applying function is written in 3 different ways

All are pronounced ‘ after’ and mean ‘do followed by ’

Examples:

(i) Evaluate

(ii) Evaluate

(iii) Find the values for for which

(iv) Find

The number of people who visit a circus can be modelled by where represents theattendance of the circus days after it opens. The profit made by the circus can be modelled by where represents the profit in euros for the circus on a day when people attend

(i) Find the number of people who were in attendance on the fourth day

(ii) Find the profit made on the fourth day

(iii) Use these functions to find a new function that will give the profit made by the circus days after the circus opens

(iv) How much profit does the circus make on the third day

Graphing functions:

To graph a function, find points which satisfy the function by substituting values in for x (inputs) and finding the corresponding y values (outputs). Plot these points and join them up to obtain the graph of the function

Vertical line test:

If any imaginary vertical line cuts the graph at only one point then it is a function.

If any imaginary vertical line cuts the graph