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 yB. 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 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
x 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 -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 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 = x
so R-1 = ( x, y ) : y = 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 the horizontal line test to determine if a function has an inverse function.
If ANY horizontal line intersects your original function in ONLY ONE location, your
function has an inverse which is also a function.
The function y= 3x+ 2, shown at the right, HAS an inverse function because it
passes the horizontal line test.
TOPIC TWO

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
a. It is not a function since 3 and 6 remain unmapped.
b. It is not a function because 2 has two images ( 5 and 6)
c. 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 ?

a. { ( x , y ) : x < y }
b. { ( x , y ) : x > y}
c. { ( 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 the range 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= 3x3-5 = 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
a. f(x) = 2x + 7 for 2 x 5
b. f(x) = x 1 for -4 x 6
c. f(x) = 5 - 3x such that -2 f(x) < 8

2. for each of the following functions state the domain and range
a. f(x) = x2
b. f(x) = x2+2
c. f(x) = 2x + 1
d. f(x) = 1 x2
Exercise 3
1.The range of the function

f(x) 3 2x for 0 0 x 7 is;

a. y: -18 y 3
b. y: -3 y 18
c. y: 3 y 18
d. 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?

a. x: - 3 x 17
b. x: - 2 x 8
c. x: -17 x 3
3.Which of the following relations represents a function:

a. R = (x, y) : y = for x 0
b. R= (x, y) : y2 = x-2 for x 0
c. R= (x, y) : y = for x 0 and y 0
d. R= (x, y) : x = 7 for all values of y
4.Which of the following relations is a function:
a. R = (x, y): -2 x 6, 3 y<8 and x<y, Where both x and y are integers
b. R= (x, y): -2 x 6, 3 y<8 and x<y, Where both x and y are integers
c. R= (x,y): y = (x+2) for x -2.
d. R = (x, y): y=(2-x) for x 2 and y 0
5.Let f (x) = x2 + 1. Which of the following is true?
a. f (-2) < f (0)
b. f (3)> f (-4)
c. f (-5) = f (5)
d. 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

1. f (x) = 3x + 2
2. f (x) = x + 6
3. f (x) = x3 + 1 etc
Examples of many to one function
1. f(x) = x2 +1
2. 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)=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.
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
a. f(x) = 3x -1
b. g (x) = x2 2x -1
c. 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?
a. f(x) = 3x x2
b. g (x) = x-1
c. k(x) =x3+1
d. f(x) =x+x2+x3
e. k(x)=x4

2. Draw the graph of the following functions:

a. f(x) = 3x x2
b. h (x) = x+1
c. g(x) =x 3- x 2+3
3. At what values of x does the graph of the function f(x) = x 2+x-6 cross thex- axis?
a. x=-3 and x=7
b. x=8 and x=-6
c. x=-3 and x=2
d. x=4 and x=-1
4. Which of the following function is one to one function?

a. f(x)=x2+2
b. f(x) =x4-x2
c. f(x)=x5-7
d. 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 x0

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

Solution
table of values.
Graph
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;
a. F(x) = 3x-6
b. F(x) =x3
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) = x-5 and then sketch the graph of f-1(x), also
state the domain and range of f-1(x).
Solution:
Graph

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) = x2-2[x] +3, what is the value of f (-4)?
Mathematics Form 3 Topics
Relations
Functions
Statistics
Rates And Variations
Sequence And Series
Circles
The Earth As A Sphere
Accounts