Download Functions - Cihan University

College of Engineering
MATHEMATICS I
Mappings and Functions
Dr Fuad M. Shareef
In this session we:
• Explain what is meant by a mapping and a
function
• Use different terms associated with function
• Express mapping and functions in different
ways
• Identify different types (simple) of
functions and draw their graphs
Why fly to Erbil International
airport in January?
Several people arriving
at Erbil International
airport were asked the
main purpose of their
visit. Their answers
were recorded:
People
Names
Azad
Jwan
Jonathan
Khalid
Shamal
Paul
Karen
Purpose of
visit
Skiing
Returning home
To study abroad
Business
This is an example of a Mapping
A mapping is any rule ( relation) which
associate two sets of items.
In this example,
• each of the names is an object, or input.
• each of the the reasons on the right is an
image, or output.
• The set of possible inputs (in our example, all
of the people who flew to Erbil International
(in January) is called the domain of the
mapping.
• The set of possible outputs (in our example,
the set of all possible reasons for flying to
Erbil (in January) is called the co-domain of
the mapping.
• The seven people questioned in this example gave
a set of four reasons, or outputs. These form the
range of the mapping for this particular inputs.
Object / Input
Image / Output
co-domain.
Domain
mapping
range
The range of any mapping forms part or all of its
co-domain.
There are four possible type of mappings:
1. one-to-one
2. one-to-many
3. many-to-many
4. many-to-one.
Here are some examples
Type of mapping
Employee
Ward
Student
drives
holds
attends
Co. Car
(One-to-one)
Patient
(One-to-many
Course
(Many-to-many)
attends
Student
Personal tutor
Many- to-one
In mathematics, many (but not all) mappings
can be expressed using algebra.
Here are some examples
Example 1
Objects
Images
-1
0
1
2
3
x
3
5
7
9
11
2x+5
Domain: Integers
General rule
Co-domain: real numbers
Example 2
Objects
Images
1.9
2.1
2.23
2.52
2.99
2
3

Domain: Integers
Co-domain: real numbers
General Rule
Rounded whole numbers
Unrounded numbers
Example 3
Objects
Images
0
1
2
3
x2  4 x  3  0
x2  x  0
x 2  3x  2  0
Domain: quadratic equation
with real roots
Co-domain: real numbers
General Rule
ax 2  bx  c  0
b 
b 2  4ac
2a
b 
b 2  4ac
2a
Discussion
For each of the previous examples :
• Decide whether the mapping is one-to-one,
one-to-many,many-to-one or many-to-many
• Take a different set of inputs and identify
the corresponding range.
What is a Function?
• Mappings which are one-to-one or
many-to-one are of particular
importance, since in these cases
there is only one possible output
for any input. Mapping of these
types are called FUNCTIONS.
• For example, in the Erbil International
airport mapping, each person gave only one
reason for the trip, but the same reason was
given by several people. This mapping is a
many-to-one mapping, so it is a function.
• The mapping in example 2, rounded whole
number onto unrounded number is NOT a
function, since, for example, the rounded
number 3 could mean any number between
2.5 and 3.4.
Questions
• Describe each of the following mappings as
either one-to-one, one-to-many, many-toone or many-to-many.
• Say whether it represents a function.
• In each case state whether the co-domain
and range are equal.
(a)
(b)
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
(d)
(c)
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
Functions & notations
• There are several different but equivalent
ways of writing down a function.
• For example, the function which maps X
onto X2 can be written in any of the following
ways.
yx
2
f ( x)  x
2
f :xx
The first two methods are commonly used.
2
Functions and their graphs
• It is useful to represent a function
graphically. For example, to draw the graph
of the function y = f(x)=2x+1, or simply
y=2x+1.
• First think about constructing a table
recording the relationship between input
and output values.
Input, x
output, f(x)
-1
-1
0
1
1
3
2
5
3
7
• Each pair of input and output values represents a single
point plotted on the graph.
• A general point is usually labelled as (x,y)
• The values of x and y are called the coordinates of the
point.
• Second, draw a pair of perpendicular axis (real number
lines) intersecting at 0 (the origin). A horizontal axis
for the input (x) and a vertical for the output y.
y=2x+1
x
-1
0
1
2
3
4
y
-1
1
3
5
7
9
• Here are some more examples:
y
y
x
5
y  x
y   ( x 2  25)
y
x
y  x3  x
For each of the above mapping, say whether it represents a
Function, and why?
x
Summary
• Explain the meaning of the terms mapping
and function.
• Explain the meaning of the terms domain,
co-domain and range of functions
• Used notation of functions.
• Draw the graph of simple (linear) function.
Coursework
Check the courses website