TOPIC 1: RELATIONS
Normally
relation deals with matching of elements from the first set called
DOMAIN with the element of the second set called RANGE.
relation deals with matching of elements from the first set called
DOMAIN with the element of the second set called RANGE.
Relations
A relation “R” is the rule that connects or links the elements of one set with the elements of the other set.
Some examples of relations are listed below:
 “Is a brother of “
 “Is a sister of “
 “Is a husband of “
 “Is equal to “
 “Is greater than “
 “Is less than “
Normally
relations between two sets are indicated by an arrow coming from one
element of the first set going to the element of the other set.
relations between two sets are indicated by an arrow coming from one
element of the first set going to the element of the other set.
Relations Between Two Sets
Find relations between two sets
The relation can be denoted as:
R = {(a, b): a is an element of the first set, b is an element of the second set}
Consider the following table
This
is the relation which can be written as a set of ordered pairs {(3,
6), (0.5, 1), (1, 2), (2, 4), (5, 10), (6, 12)}. The table shows that
the relation satisfies the equation y=2x. The relation R defining the
set of all ordered pairs (x, y) such that y = 2x can be written
symbolically as:
is the relation which can be written as a set of ordered pairs {(3,
6), (0.5, 1), (1, 2), (2, 4), (5, 10), (6, 12)}. The table shows that
the relation satisfies the equation y=2x. The relation R defining the
set of all ordered pairs (x, y) such that y = 2x can be written
symbolically as:
R = {(x, y): y = 2X}.
Relations Between Members in a Set
Find relations between members in a set
Which of the following ordered pairs belong to the relation {(x, y): y>x}?
(1, 2), (2, 1), (3, 4), (3, 5), (2, 2), (8, 0), (8, 3).
Solution.
(1, 2), (3, 4), (8, 0), (8,3).
Relations Pictorially
Demonstrate relations pictorially
For
example the relation ” is greater than ” involving numbers 1,2,3,4,5
and 6 where 1,3 and 5 belong to set A and 2,4 and 6 belong to set B can
be indicate as follows:
example the relation ” is greater than ” involving numbers 1,2,3,4,5
and 6 where 1,3 and 5 belong to set A and 2,4 and 6 belong to set B can
be indicate as follows:
This kind of relation representation is referred to as pictorial representation.
Relations
can also be defined in terms of ordered pairs (a,b) for which a is
related to b and a is an element of set A while b is an element of set
B.
can also be defined in terms of ordered pairs (a,b) for which a is
related to b and a is an element of set A while b is an element of set
B.
For
example the relation ” is a factor of ” for numbers 2,3,5,6,7 and 10
where 2,3,5 and 6 belong to set A and 6,7 and 10 belong to set B can be
illustrated as follows:
example the relation ” is a factor of ” for numbers 2,3,5,6,7 and 10
where 2,3,5 and 6 belong to set A and 6,7 and 10 belong to set B can be
illustrated as follows:
Example 1
<!–
[if !supportLists]–>1. Draw an arrow diagram to illustrate the
relation which connects each element of set A with its square.
[if !supportLists]–>1. Draw an arrow diagram to illustrate the
relation which connects each element of set A with its square.
Solution
Example 2
Using
the information given in example 1, write down the relation in set
notation of ordered pairs. List the elements of ordered pairs.
the information given in example 1, write down the relation in set
notation of ordered pairs. List the elements of ordered pairs.
Example 3
As we,
Solution;
Example 4
Let X= {2, 3, 4 } and Y= {3 ,4, 5}
Draw an arrow diagram to illustrate the relation ” is less than”
Exercise 1
Let P= {Tanzania, China, Burundi, Nigeria}
Draw a pictorial diagram between P and itself to show the relation
“Has a larger population than”
2. Let A = 9,10,14,12 and B = 2,5,7,9 Draw an arrow diagram between A and B to illustrate the relation ” is a multiple of”
3.Let A = mass, Length, time and
B = {Centimeters, Seconds, Hours, Kilograms, Tones}
Use the set notation of ordered pairs to illustrate the relation “Can be measured in”
4.
A group people contain the following; Paul Koko, Alice Juma, Paul
Hassan and Musa Koko. Let F be the set of all first names, and S the set
of all second names.
A group people contain the following; Paul Koko, Alice Juma, Paul
Hassan and Musa Koko. Let F be the set of all first names, and S the set
of all second names.
Draw an arrow diagram to show the connection between F and S
5. Let R={ (x, y): y=x+2}
Where x∈A and A ={ 1,0,1,2}
and y∈B, List all members of set B
Exercise 2
1. Let the relation be defined
Consider the following pictorial diagram representing a relation R.
Let the relation R be defined as
A relation R on sets a and B where A = 1,2,3,4,5 and B = 7,8,9,10,11,12 is defined as ” is a factor of “
Graph of a Relation
A Graph of a Relation Represented by a Linear Inequality
Draw a graph of a relation represented by a linear inequality
Given
a relation between two sets of numbers, a graph of the relation is
obtained by plotting all the ordered pairs of numbers which occur in the
relation
a relation between two sets of numbers, a graph of the relation is
obtained by plotting all the ordered pairs of numbers which occur in the
relation
Consider the following relation
The graph of R is shown the following diagram( xy plane).
Example 5
Solved:
Note that some relations have graphs representing special figures like straight lines or curves.
Example 6
Draw the graph for the relation R= {(x, y): y = 2x +1} Where both x and y are real numbers.
Solution
The
equation y = 2x +1 represents a straight line, this line passes throng
uncountable points. To draw its graph we must have at least two points
through which the line passes.
equation y = 2x +1 represents a straight line, this line passes throng
uncountable points. To draw its graph we must have at least two points
through which the line passes.
Graph;
Example 7
Let A = {2,1,0, 1, 2 } and B ={0,1,2,3,4}
Let the relation R be y= x^{2, }where x ∈A and y∈B. Draw the graph of R
Solution
NB:
When the relation is given by an equation such as y = f (x), the domain
is the set containing x values satisfying the equation and the range
is the set of yvalues satisfying the given equation.
When the relation is given by an equation such as y = f (x), the domain
is the set containing x values satisfying the equation and the range
is the set of yvalues satisfying the given equation.
Exercise 3
Test Yourself:
Quiz.
Domain and Range of a Relation
The Domain of Relation
State the domain of relation
Domain:
The domain of a function is the set of all possible input values (often
the “x” variable), which produce a valid output from a particular
function. It is the set of all real numbers for which a function is
mathematically defined.
The domain of a function is the set of all possible input values (often
the “x” variable), which produce a valid output from a particular
function. It is the set of all real numbers for which a function is
mathematically defined.
The Range of a Relation
State the range of a relation
Range:
The range is the set of all possible output values (usually the
variable y, or sometimes expressed as f(x)), which result from using a
particular function.
The range is the set of all possible output values (usually the
variable y, or sometimes expressed as f(x)), which result from using a
particular function.
If
R is the relation on two sets A and B such that set A is an independent
set while B is the dependent set, then set A is the Domain while B is
the Codomain or Range.
R is the relation on two sets A and B such that set A is an independent
set while B is the dependent set, then set A is the Domain while B is
the Codomain or Range.
Note
that each member of set A must be mapped to at least one element of set
B and each member of set B must be an image of at least one element in
set A.
that each member of set A must be mapped to at least one element of set
B and each member of set B must be an image of at least one element in
set A.
Consider the following relation
Example 8
Let P = 1,3,4,10 and Q = 0,4,8
Find the domain and range of the relation R:” is less than”
Example 9
As we,
Exercise 4
1. Let A = { 3,5,7,9 } and B = {1,4,6,8 } , find the domain and range of the relation “is greater than on sets A and B
4. Let X ={3, 4, 5, 6} and
Y ={2, 4, 6, 8}
Draw the pictorial diagram to illustrate the relation “is less than or equal to‘ and state its domain and range
Inequalities:
The equations involving the signs < , ≤, > or ³ are called inequalities
Eg. x<3 x is less than 3
x>3 x is greater than 3
x≤ 2 x is less or equal to 2
x³ 2 x is greater or equal to 2
x > y x is greater or than y etc
Inequalities can be shown on a number line as in the following
Inequalities involving two variables:
If
the inequality involves two variables it is treated as an equation and
its graph is drawn in such a way that a dotted line is used for > and
< signs while normal lines are used for those involving ≤ and ≥.
the inequality involves two variables it is treated as an equation and
its graph is drawn in such a way that a dotted line is used for > and
< signs while normal lines are used for those involving ≤ and ≥.
The line drawn separates the xy plane into two parts/regions
The
region satisfying the given inequality is shaded and before shading it
must be tested by choosing one point lying in any of the two regions,
region satisfying the given inequality is shaded and before shading it
must be tested by choosing one point lying in any of the two regions,
Example 10
1. Draw the graph of the relation R = {(x, y): x>y}
Solution:
x>y is the line x =y but a dotted line is used.
Graph
If you draw a graph of the relation R = {(x,y ) : x < y} , the same line is draw but shading is done on the upper part of the line.
Exercise 5
1. Draw the graph of the relation R = {(x,y ): x + y > 0}
2 .Draw the graph of the relation R = {( x ,y ) : x – y ³ 2}
3. Write down the inequality for the relation given by the following graph
4. Draw a graph of the inequality for the relation x >2 and shade the required region.
Domain and Range from the graph
Definition: Domain is the set of all x values that satisfy the given equation or inequality.
Similarly Range is the set of all y value satisfying the given equation or inequality
Example 11
1. Consider the following graph and state its domain and range.
Solution
Example 12
State the domain and range of the relation whose graph is given below.
Inverse of a Relation
The Inverse of a Relation Pictorially
Explain the Inverse of a relation pictorially
If there is a relation between two sets A and B interchanging A and B gives the inverse of the relation.
If R is the relation, then its inverse is denoted by R^{1}
 If the relation is shown by an arrow diagram then reversing the direction of the arrow gives its inverse

If the relation is given by ordered pair ( x, y) , then inter changing
the variables gives inverse of the relation, that is (y,x) is the
inverse of the relation. So domain of R = Range of R 1 and range of R =
domain of R^{1}
Example 13
1.
The inverse of this relation is “ is a multiple of “
Inverse of a Relation
Find inverse of a relation
Example 14
Find the inverse of the relation R ={ ( x, y):x+ 3 ³ y}
Solution
R^{1} is obtained by inter changing the variables x and y.
Example 15
Find the inverse of the relation
R ={ ( x , y ): y = 2x }
Solution
R ={( x , y ): y = 2x }
After interchanging the variable x and y, the equation
y = 2x becomes x = 2y
or y = ½ x
so R^{1} = ( x, y ) : y = ½ x
Exercise 6
1
.Let A = 3,4,5 and B ‘= 1,4,7 find the inverse of the reaction “ is
less than “ which maps an element from set A on to the element in set B
.Let A = 3,4,5 and B ‘= 1,4,7 find the inverse of the reaction “ is
less than “ which maps an element from set A on to the element in set B
2 .Find the inverse of the relation R = {( x ,y ) : y > x – 1}
3 .Find the inverse of the following relation represented in pictorial diagram
4 .State the domain and range for the relation given in question 3 above
5. State the domain and range of the inverse of the relation given in question 1 above.
A Graph of the Inverse of a Relation
Draw a graph of the inverse of a relation
Use thehorizontal line testto determine if a function has aninverse function.
If
ANY horizontal line intersects your original function in ONLY ONE
location, your function has an inverse which is also a function.
ANY horizontal line intersects your original function in ONLY ONE
location, your function has an inverse which is also a function.
The functiony= 3x+ 2, shown at the right, HAS aninverse functionbecause it passes the horizontal line test.
TOPIC 2: FUNCTIONS
Normally
relation deals with matching of elements from the first set called
DOMAIN with the element of the second set called RANGE.
relation deals with matching of elements from the first set called
DOMAIN with the element of the second set called RANGE.
Definitions:
A
function is a relation with a property that for each element in the
domain there is only one corresponding element in the range or co
domain
function is a relation with a property that for each element in the
domain there is only one corresponding element in the range or co
domain
Therefore functions are relations but not all relations are functions
Representation of a Function
The Concept of a Functions Pictorially
Explain the concept of a functions pictorially
Example 1
Which of the following relation are functions?
Solution
 It is not a function since 3 and 6 remain unmapped.
 It is not a function because 2 has two images ( 5 and 6)
 It is a function because each of 1, 2, 3 and 4 is connected to exactly one of 5, 6 or 7.
Functions
Identify functions
TESTING FOR FUNCTIONS;
If
a line parallel to the yaxis is drawn and it passes through two or
more points on the graph of the relation then the relation is not a
function.
a line parallel to the yaxis is drawn and it passes through two or
more points on the graph of the relation then the relation is not a
function.
If it passes through only one point then the relation is a function
Example 2
Identify each of the following graphs as functions or not.
Exercise 1
1. Which of the following relations are functions?
2. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
and B ={ 2, 3, 5, 7 }
Draw an arrow diagram to illustrate the relation “ is a multiple of ‘ is it a function ? why?
3. let A = {1,1 ,2,2} and
B = {1, 2, 3, 4 } which of the following relations are functions ?
 { ( x , y ) : x < y }
 { ( x , y ) : x > y}
 { ( x , y ) : y = x^{2}}
Domain and Range of a Function
The Domain of a Function
State the domain of a function
If y = f (x),that is y is a function of x ,then domain is a set of x values that satisfy the equation y = f (x).
The Range of Function
State the range of function
If y = f (x),that is y is a function of x , then therange is a set of y value satisfying the equation y = f (x).
Example 3
1. Let f(x) = 3x – 5 for all value of x such that 2 £ x £ 3 find its range
Solution
f (x) = y = 3x 5
When x = 2
f(2) = y = 3x(2)5 = 11 , so (x,y)=(2,11)
f(3) =y= 3×35 = 4, so when x = 3 , y = 4
Therefore y is found in between – 11 and 4
Range ={ y: – 11 £ y £ 4}
Example 4
If f (x) = x^{2 }– 3, state the domain and range of f (x)
Solutions;
Domain = all real numbers
Range:
f(x) = y = x^{2} – 3
Make x the subject
y+ 3 = x^{2}
Exercise 2
1. For each of the following functions, state the domain and range
 f(x) = 2x + 7 for 2 £ x £ 5
 f(x) = x – 1 for 4 £ x £ 6
 f(x) = 5 – 3x such that 2 £ f(x) < 8
2. for each of the following functions state the domain and range
 f(x) = x^{2}
 f(x) = x^{2}+2
 f(x) = 2x + 1
 f(x) = 1 – x^{2}
Exercise 3
1.The range of the function
f(x) 3 – 2x for 0 ³0 x £7 is;
 y: 18£ y £3
 y: 3£ y £18
 y: 3 £y £18
 y: 18 £ y £3
2. The range of the function
f(x)=2x+1 is y: 3£ y £17 what is the domain of this function?
 x: – 3£ x £17
 x: – 2£ x £8
 x: 17 £ x £3
3.Which of the following relations represents a function:
 R = (x, y) : y = for x ≥0
 R= (x, y) : y^{2 }= x2 for x ≥0
 R= (x, y) : y = for x ≥0 and y ≥0
 R= (x, y) : x = 7 for all values of y
4.Which of the following relations is a function:
 R = (x, y): 2 £ x £6, 3 £ y<8 and x<y, Where both x and y are integers
 R= (x, y): 2 £ x £6, 3 £ y<8 and x<y, Where both x and y are integers
 R= (x,y): y = √(x+2) for x ≥2.
 R = (x, y): y=√(2x) for x ≤2 and y ≤0
5.Let f (x) = x^{2 }+ 1. Which of the following is true?
 f (2) < f (0)
 f (3)> f (4)
 f (5) = f (5)
 The function crosses , y – axis at 1
One to one and many to one functions:
One to functions;
A one to one function is a function in which one element from the domain is mapped to exactly one element in the range:
That is if a ≠b then f (a) ≠f (b)
Many to one function;
This is another type of function with a property that two or more elements from the domain can have one image (the same image).
Examples of one to one functions
 f (x) = 3x + 2
 f (x) = x + 6
 f (x) = x^{3} + 1 etc
Examples of many to one function
 f(x) = x^{2} +1
 f(x) = x^{4} – 2 etc
NB. All functions with odd degrees are one to one function and all functions with even degrees are many to one functions.
Example 5
Let A = 2, 1, 0, 1, 2 and B = 0, 1, 4 and the function f mapping each element from set A to those of B is defined as f(x)=x^{2}.Is f one to one function?
Example 6
Let P = {2, 1, 0,1,2} and
Q = {1, 0, 1, 2, 3}
g(x) = x + 1, is g one to one function?
Solution:
g (x) is one to one function because every element in P has only one image in Q
NB:
In example 1, f(x) is not a one to one function because 2 and 2 in A
have the same image in B, that is 4 is the image of both 2 and 2.
In example 1, f(x) is not a one to one function because 2 and 2 in A
have the same image in B, that is 4 is the image of both 2 and 2.
Also 1 is the image of both 1 nd 1.
Example 7
State whether or not if the following graphs represent a one to one function:
Solution:
Draw
a line parallel to the x axis and see if it crosses the graph at more
than one points. If it does, then, the function is many to one and if it
crosses at only one point then the graph represents a one to one
function.
a line parallel to the x axis and see if it crosses the graph at more
than one points. If it does, then, the function is many to one and if it
crosses at only one point then the graph represents a one to one
function.
Graphic Function
Graphs of Functions
Draw graphs of functions
Many
functions are given as equations, this being the case, drawing a graph
of the equation is obtaining the graph of the equation which defines the
function.
functions are given as equations, this being the case, drawing a graph
of the equation is obtaining the graph of the equation which defines the
function.
Note
that, you can draw a graph of a function if you know the limits of its
independent variables as well as dependent variables. i.e you must know
the domain and range of the given function.
that, you can draw a graph of a function if you know the limits of its
independent variables as well as dependent variables. i.e you must know
the domain and range of the given function.
Example 8
Draw the graph of the following functions
 f(x) = 3x 1
 g (x) = x^{2} – 2x 1
 h (x) = x^{3}
Solution
f(x) = 3x – 1
The domain and range of f are the sets of all real numbers
f(x) = y = 3x – 1
So y = 3x – 1
Table of value :
g(x) = x^{2} 2x 1
y=x^{2}21
a=1, b=2 1 and c=1
forh(x) = x^{3}
Solution:
The
first graph is the graph of linear function, the second one is called
the graph of a quadratic function and the last graph is for cubic
function.
first graph is the graph of linear function, the second one is called
the graph of a quadratic function and the last graph is for cubic
function.
Example 9
Draw a graph of the function:
f(x) = 1 + 6xx^{2}
Solution:
a=1, b=6, c=1
Exercise 4
1.Which of the following are one to one function?
 f(x) = 3x – x^{2}
 g (x) = x1
 k(x) =x^{3}+1
 f(x) =x+x^{2}+x^{3}
 k(x)=x^{4}
2. Draw the graph of the following functions:
 f(x) = 3x – x2
 h (x) = x+1
 g(x) =x ^{3}– x ^{2}+3
3. At what values of x does the graph of the function f(x) = x^{2}+x6 cross thex axis?
 x=3 and x=7
 x=8 and x=6
 x=3 and x=2
 x=4 and x=1
4. Which of the following function is one to one function?
 f(x)=x^{2}+2
 f(x) =x^{4}x^{2}
 f(x)=x^{5}7
 f(x)=x^{2}+x+2
Functions with more than one part.
Some functions consist of more than one part. When drawing their graphs draw the parts separately.
If
the graph includes an end point, indicate it with a solid dot if it
does not include the end point indicate it with a hollow dot.
the graph includes an end point, indicate it with a solid dot if it
does not include the end point indicate it with a hollow dot.
E.g. draw the graphs of the functions
(a) F(x) x+1 for x>0
(b) f(x)=x+1for x³0
Example 10
Solved.
(c) Sketch its graph
(d) State the domain and range of f
Solution:
Exercise 5
Sketch the graph of each of the following functions and for each case state the domain and range.
Absolute value functions (Modulus functions)
The absolute function is defined
Table of values
Example 11
Solve the following <!–[endif]–>
Solution
table of values.
Step functions:
Example 12
Draw the graph of
Note that the graph obtained is like steps such functions are called steps functions
Exercise 6
1. Draw the graph of
Inverse of a Function
The Inverse of a Function
Explain the inverse of a function
In the discussion about relation we defined the inverse of relation.
It is true that the inverse of the relation is also a relation.
Similarly because a function is also relation then every function has its inverse
The Inverse of a Function Pictorially
Show the inverse of a function pictorially
According to the definition of function the inverse of a function is also a function if and only if the function is one to one
The Inverse of a Function
Find the inverse of a function
If the function f is one to one function given by an equation, then its inverse is denoted by f^{1} which is obtained by inter changing the variables x and y then making y the subject of the formula.
I.e. If y=f(x), then x = f^{1} (y)
Example 13
1. Find the inverse of each of the following functions;
 F(x) = 3x6
 F(x) =x^{3}
Solution:
A Graph of the Inverse of a Function
Draw a graph of the inverse of a function
Example 14
find the inverse of the function f(x) = x5 and then sketch the graph of f^{1}(x) , also state the domain and range of f^{1}(x).
solution:
Domain = {All real numbers}
Range = {All real numbers}
NB: if a function f takes a domain A to a range B, then the inverse f^{1} takes B back to A.
Hence the domain of f^{1} is the range of f, and the range of f^{1} is the domain of f.
The Domain and Range of Inverse of Functions
State the domain and range of inverse of functions
Example 15
Solve;
Solutions:
Exercise 7
1.Find the inverse of each of the following functions:
Exercise 8
1. given that f(x) = x^{2}2^{[x]} +3, what is the value of f (4)?
MATHEMATICS FORM THREE OTHER TOPICS
FORM THREE MATHEMATICS STUDY NOTES TOPIC 12.
FORM THREE MATHEMATICS STUDY NOTES TOPIC 34.
FORM THREE MATHEMATICS STUDY NOTES TOPIC 5.
FORM THREE MATHEMTICS STUDY NOTES TOPIC 67.
FORM THREE MATHEMTICS STUDY NOTES TOPIC 8.
O’LEVEL MATHEMATICS NOTES
FORM ONE MATHEMATICS STUDY NOTES
FORM TWO MATHEMATICS STUDY NOTES
FORM THREE MATHEMATICS STUDY NOTES
FORM FOUR MATHEMATICS STUDY NOTES