Home News FORM THREE MATHEMATICS STUDY NOTES TOPIC 1-2.

FORM THREE MATHEMATICS STUDY NOTES TOPIC 1-2.

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.

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

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:

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:-

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.

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 1

<!–
[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.

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.

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

Consider the following relation

The graph of R is shown the following diagram( x-y 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.

Graph;

Example 7

Let A = {-2,-1,0, 1, 2 } and B ={0,1,2,3,4}

Let the relation R be y= x2, 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 y-values 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 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.

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 Co-domain 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.

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 line drawn separates the x-y 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,

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

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

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.

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.

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

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 y-axis 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 = x2}

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= 3×3-5 = 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) = x2 – 3, state the domain and range of f (x)

Solutions;

Domain = all real numbers

Range:

f(x) = y = x2 – 3

Make x the subject

y+ 3 = x2

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) = x2
f(x) = x2+2
f(x) = 2x + 1
f(x) = 1 – x2

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) : y2 = x-2 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=√(2-x) for x ≤2 and y ≤0

5.Let f (x) = x2 + 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) = x3 + 1 etc

Examples of many to one function

f(x) = x2 +1
f(x) = x4 – 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)=x2.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.

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.

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.

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.

Example 8

Draw the graph of the following functions

f(x) = 3x -1
g (x) = x2 – 2x -1
h (x) = x3

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) = x2 -2x -1

y=x2-2-1

a=-1, b=-2 1 and c=-1

forh(x) = x3

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.

Example 9

Draw a graph of the function:

f(x) = -1 + 6x-x2

Solution:

a=-1, b=6, c=-1

Exercise 4

1.Which of the following are one to one function?

f(x) = 3x – x2
g (x) = x-1
k(x) =x3+1
f(x) =x+x2+x3
k(x)=x4

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) = x2+x-6 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)=x2+2
f(x) =x4-x2
f(x)=x5-7
f(x)=x2+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.

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.

(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;