MATHS METHODS
FUNCTIONS, RELATIONS AND TRANSFORMATIONS
SETS
SET NOTATION



A set is a collection of numbers or objects
Every number or object in the set is called an element
There are a series of symbols that you need to have a working knowledge of in relation to sets:
symbol
Definition
example
is an element of
4
{even numbers}
is not an element of
8
{2, 3, 4, 7}
A=B
A is equal to B only if all elements of the set are exactly the
same
A⇔B
A and B are equivalent if they have the same number of
elements
A = {odd numbers less than 10}
B = {1, 3, 5, 7, 9} A = B
A = {3, 7, 10} and B = {4, 21, 25}
A⇔B
Empty or null set, has no elements
or U
B
Universal set – all elements which are being considered
A
B is a subset of A
A\B
Elements of A that are not elements of B
A
B
Union of A and B - set of elements in A or B or both
A
B
Intersection of A and B – set of elements in both A and B
A = {2, 4, 6, 8} and B = {2, 6}
B A
A = {2, 4, 6, 8} and B = {2, 6}
A \ B = {4, 8}
The complement of A is all members of the universal set which
are not elements of A
A’
Finite set
The number of elements can be counted
Infinite set
The number of elements cannot be specified
NUMBER SYSTEMS


All numbers can be categorised into varying sets depending on their properties
The key number sets you need to be familiar with are:
-
– real numbers. All numbers, both rational and irrational (eg 4, 7, π, , -10)
-
– rational numbers. Numbers of the form , where p and q are non-zero integers
-
– integers. Whole numbers, {...-2, -1, 0, 1, 2, ...}
– natural numbers. Positive integers, {0, 1, 2...}
INTERVALS

Interval notation indicates whether a number is to be included or not
- A square bracket [ and a closed circle  indicate a number is included
- A round bracket ( and an open circle ◦ indicate a number is excluded
For example: {
}
(
]
-1
RELATIONS, DOMAIN AND RANGE


A relation is a set of ordered pairs, (
. Every relation determines two sets:
- Domain – the set of values that x may have
- Range – the set of values that y may have
A relation may be a set of discrete points that can be listed, or it may be continuous.
5
MAPPINGS

A mapping consists of 2 sets (the domain and range) and a rule for assigning to each element in the first set
one or more elements in the second set

We say that A is mapped to B and write this as m : A  B
IMPLIED DOMAIN


Also known as maximal domain
When no domain is stipulated, we assume the domain is taken from the largest for which the rule has
meaning
maximal domain is
]
- (
maximal domain is [
Exercise 6B
FUNCTIONS


A function is a relation for which each x value has a single and unique y value
- No two ordered pairs have the same first element
The vertical line test is one way of identifying functions
For example: of the two relations below, which one would be considered a function?

There are four types of relations (NOTE only two of these are considered functions)
- One-one functions have only one x value for every y value (function)
- Many-one functions have more than one x value which corresponds to a particular y value (function)
- One-many relations have more than one y value which corresponds to a particular x value
- Many-many relations have more than one x value corresponding with more than one y value
For example: Graph A is a one-one function, where as Graph B is a one-many function
FUNCTION NOTATION


Usually denoted with a lower case letter – f, g, h
Various notation is used across different textbooks and across the world
(

We use this one most commonly, read as f of x
We can also incorporate a restricted domain in our function notation
For example:
[
]
indicates the graph of
is drawn for values of x such that

Note: the notation [ ]
or in general terms
, X is the domain, but Y is not necessarily the
range. Y is the set that contains the range and is called the co-domain.

Using function notation we can show the substitution of values of x into the function
For example: For the function (
, find ( , (
, (
and (
RESTRICTING A FUNCTION
Consider the following:
(
(
(
Exercise 6C, 6D
HYBRID FUNCTIONS


When a function has different rules for different subsets of the domain, also known as step functions
May have previously covered in the form of segmented line graphs (linear)
For example:

(
{
To sketch a hybrid function, consider the end points first
Exercise 6E
INVERSE FUNCTIONS

If a function, f, is a one-to-one function then it’s inverse is given as
(

, for
ran ,
dom
} (graphed in green below)
(
(
(
(
Consider the relation {(
- The inverse of this relation is found by interchanging the coordinates of each ordered pair in the
relation
} (graphed in blue below)
(
(
(
(
- The inverse relation is {(
- It is easy to see from the graphs that the inverse
of the function is the original reflected in the line
-
The x values of the original relation have become
the y values of the inverse, just as the y values of
the original relation have become the x values of
the inverse
That is, the domain of the original has become the
range of the inverse, and the range of the original
has become the domain of the inverse
-

if (
NOTE: a function only has an inverse if it is a one-to-one function
For example: try sketching the graph of the inverse in each of the following
Find the inverse function of each and state the domain and range of
a
(
]
(
b
[
(
interchange the x and y, and then
solve for y
√
c
(
(
√
(
Given ran f = dom
(
[
√
(
Range = (
Domain = (
]
Range = [
]
(
√
Domain = [
Range = R
Domain = [
TRANSFORMATIONS OF FUNCTIONS



A transformation of points in the x – y plane occurs when every point in the plane is given a new position
according to some rule
A convenient method of defining this new position is to represent each ordered pair (
of a function as
transformed to a new ordered pair (
(read x-dash, y-dash) according to a rule
Now translations can be defined as follows:
(
(
(
Read as: Original point changes to new point according to rule
TRANSLATION

Consider the following:
-
How has each function changed? What transformation has occurred?
-
Can we represent this using the notation above?
In general:
Type of transformation
Movement
Horizontal translation
h units to the right
(
(
(
Horizontal translation
h units to the left
(
(
(
Vertical translation
k units up
(
(
(
Vertical translation
k units down
(
(
(
For example: describe the transformations that have taken place if:
(
(
(
b) (
(
(
a)
Notation
(
(
(
d) (
(
(
c)

( if the transformation is described by a horizontal
To find the rule of the transformed function, say
translation of h units to the right followed by a vertical translation of k units up
(
1 Convert to (
notation
(
(
(
2 Form two equations equating the first coordinates with the second coordinates (hint: start with
)
3 Solve for y and x in terms of y’ and x’
4 Substitute y and x back into the original function given (
( )
(
(
5 Drop the dashes and solve for y
(
(
For example: if
1 (
is transformed 1 unit to the right and 2 units up:
(
2
3
4
(
5
(
(
(
This is now our rule for
the transformed
function
Now try these…
a)
if
is transformed 2 units to the left and 3 units down, find the rule of the transformed function
b) repeat if
c)
√
repeat if
Now complete Exercise 6H – here rules are already given, you just need to SKETCH
DILATION AND REFLECT ION


Dilations and reflections can occur both from the x-axis and y-axis
Consider the following:
-
How has each function changed?
-
Can we represent this change using the function notation for transformations?
-
How has each function changed?
-
Can we represent this change using the function notation for transformations?
In general:
Type of transformation
Movement
Notation
Dilation
From x axis by factor of a
(
(
(
Dilation
From y axis by factor of a
(
(
(
Reflection
In x axis
(
(
(
Reflection
In y axis
(
(
(
For example: describe the transformations that have taken place if the transformations are defined by:
a)
(
(
(
)
hint: as a general rule, start with dilations, followed by reflections, followed by translations
a dilation of factor _____ from the _____ axis, followed by
a dilation of factor _____ from the _____ axis, followed by
a reflection in the _____ axis, followed by
a horizontal translation of _____ units to the _____, followed by
a vertical translation of _____ units _____
b) If
is the given function, find the rule of the transformed function for the transformations in a) above
1 (
(
(
2
3
(
4
(
(
(
5
(
(
)
)
c)
Do the same but with
√
d) Do the same but with
Exercise 6I Questions 1, 2
Exercise 6J Questions 1, 2
FINDING TRANSFORMATIONS FROM THE EQUATION


What if you are given the rule of the transformed function and you have to find the sequence of
transformations that took place algebraically?
We go through the same process as before, but work backwards
For example: find a sequence of transformations that take the graph of
(
6
4
3
(given)
(
(get step 5 into form
(
)
(apply dashes)
(
(NOTE x expression not squared)
2
1 (
to the graph of
( equate to y and x and solve for y’ and x’)
(
(
(define transformation of point
Now use this to describe the sequence of transformations required:
A dilation of factor 3 from the x axis
A reflection in the x axis
A translation of 1 unit to the left
A translation of 2 units down
Exercise 6J Question 3
(