1. College of Engineering MATHEMATICS I
Mappings and Functions
Dr Fuad M. Shareef

2. 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

3. 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:

4. This is an example of a Mapping
People
Names
Purpose of
visit
Azad
Skiing
Jwan
Jonathan
Returning home
Khalid
To study abroad
Shamal
Business
Paul
Karen
This is an example of a Mapping

5. 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.

6. The range of any mapping forms part or all of its
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.
Image / Output
Object / Input
Domain
co-domain.
mapping
range
The range of any mapping forms part or all of its
co-domain.

7. There are four possible type of mappings: 1. one-to-one 2
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
Co. Car
drives
(One-to-one)
Ward
Patient
holds
(One-to-many
Student
Course
attends
(Many-to-many)
attends
Student
Personal tutor
Many- to-one

8. Co-domain: real numbers
In mathematics, many (but not all) mappings can be expressed using algebra. Here are some examples
Example 1
Objects
Images -1 1 2 3 x 3 5 7 9 11
2x+5
General rule
Co-domain: real numbers
Domain: Integers

9. Co-domain: real numbers Rounded whole numbers Unrounded numbers
Example 2
Objects
Images 2 3
1.9
2.1
2.23
2.52
2.99
Domain: Integers
Co-domain: real numbers
General Rule
Rounded whole numbers Unrounded numbers

10. Co-domain: real numbers
Example 3
Images
Objects 1 2 3
Domain: quadratic equation
with real roots
Co-domain: real numbers
General Rule

11. 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.

12. 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.

13. 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.

14. Questions
Describe each of the following mappings as either one-to-one, one-to-many, many-to-one or many-to-many.
Say whether it represents a function.
In each case state whether the co-domain and range are equal.

15. (a)
(b) O O O O
(d)
(c) O O O O

16. 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.
The first two methods are commonly used.

17. 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 -1 1 2 3
output, f(x) 5 7

18. 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. x y -1 1 3 2 5 7 4 9

19. Here are some more examples:y y x x 5 y x
For each of the above mapping, say whether it represents a
Function, and why?

20. 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.

21. Coursework
Check the courses website