## Relations and Functions

relation is a connection between the elements of a set or the elements of two sets.

e.g. "is greater than" and " is twice the size of " are examples of relations.

A function is a special type of relation.

### Representation

Relations can be represented in several ways:

• As a set of ordered pairs.

e.g. A = {(1, 2), (2, 3), (3, 4)}

The first value in each ordered pair is the x-value.

The second value in each ordered pair is the y-value.

• As an arrow graph or mapping.

e.g. The domain The range

The set of x-values is called the domain. i.e. {1, 2, 3}

The set of y-values is called the range. i.e. {2, 3, 4}

• In table form

e.g.

 Domain Range x y 1 2 2 3 3 4

• On a graph.

e.g. • By a rule or formula.

e.g. y = x + 1

function is a relation where each of the members of the domain,(each x value) is connected to only one member of the range, (the y value). Most relations are also functions.

Exceptions are relations such as x 2 + y 2 = 9 which has a graph of a circle. This relation is not a function.

### Sequences

In mathematics, numbers are often arranged in a sequence. These sequences can be shown be a formula or function.

Often a sequence follows a rule or pattern.

Easy Sequence
Some sequences are obvious: 2, 4, 6, 8, 10...
This is the sequence of EVEN numbers.

The formula for this sequence is 2n where n stands for a counting number.

To find the next number in this sequence, calculate 2 × 6 = 12

Slightly Harder Sequence
The sequence 1, 4, 9, 16, 25... is the sequence of square numbers.
Comparing this sequence to the set of counting numbers {1, 2, 3, 4..}

 1 2 3 4 5 ... 1 4 9 16 25 ...

The formula for this sequence is n2.
The next number in this sequence is 62 = 36

More Difficult Sequence
The number of diagonals in polygons forms a sequence.

 Name of polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Number of sides 3 4 5 6 7 Number of diagonals 0 2 5 9 14

The sequence for the number of diagonals is 0, 2, 5, 9, 14...
The pattern for this sequence is an increase of 2, 3, 4 between each term.
The formula for this sequence is (n2 + n − 2)/2
The next number in this sequence is 20