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:

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

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 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.

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

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

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

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

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-to-one 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 (d) (c) 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. The first two methods are commonly used.

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

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

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?

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.

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