Basic Mathematics by Prof. Jairus. Khalagai - HTML preview

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Introduction

a) Story of Maize Grinding Machine

Jane walks in a village to a nearby market carrying a basket of maize to be ground

into flour. She puts the maize into a container in the grinding machine and starts

rotating the handle. The maize is then ground into flour which comes out of the

machine for her to take home.

Question

What relation can you make among the maize, the grinding machine and the

flour?

b) Story of children born on the Christmas day in the year 2005

It was reported on the 25th of December 2005 in Pumwani Maternity Hospital

which is in Nairobi the Capital City of Kenya that mothers who gave birth to

single babies were a total of 52. This was the highest tally on that occasion. As

it is always the case each baby was given a tag to identify him or her with the

mother.

Questions

1. In the situation above given the mother how do we trace the baby?

2. Given the baby how do we trace the mother?

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Activity

Note that we can now represent the story of the maize grinding machine dia-

grammatically as follows:

A

B

f

A = Set of some content (in this case maize) to be put in the grinding

machine.

f

= The mapping or function representing the process in the grinding

machine

B = Set of the product content (in this case flour) to be obtained

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Example 1

In this example we define two sets and a relation between them as follows:

Let A = {2, 3, 4}

B = {2, 4, 6, 8}

f is a relationship which says “is a factor of”

e.g. 3 is a factor of 6

In this case we have the following mapping:

A

B

2

2

4

3

6

4

8

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Example 2

Think of a number of such situations and represent them with a mapping diagram

as shown above.

In our second story of each mother giving birth to only one child can be repre-

sented in a mapping diagram as follows:

A

B

f

X

X

X

X

A = Set of babies

B = Set of mothers

f

= Relation which says “baby to”

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Remarks 3

i. Notice that in this mapping each object is mapped onto a unique image. In

this case it is a function. We write f: A→ B

ii. Note also that in the mapping above even if we interchanged the roles of sets

A and B we still have that each object has a unique image. Thus we have…

B

A

g

X

X

X

X

In this case we have

B = Set of mothers

A = Set of babies

g = Relation which says “is mother of ”

In this case we say that the function f has an inverse g. We normally denote this inverse g as f-1

Thus for f: A → B we have f-1: B → A

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Example 4

Let A = {1, 2, 3, 4, 5}

B = {2, 3, 5, 7, 9, 11, 12}

f : x → 2 x + 1

Then we have the mapping as follows: f: x → 2 x+ 1

2

1

3

5

2

7

9

3

11

12

4

5

For notation purposes in this mapping we can also write:

f(1) = 3, f(2) = 5 etc

In general f( x) = 2 x + 1

The set A is called the domain of f and the set B is called the co domain of f.

The set { 3, 5, 7, 9, 11} within 13 on which all elements of A are mapped is calle

x − 1

range of f. Note that here the inverse of f is given by −1

f (x) =

and is also

2

a function.

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Exercise 5

Starting with the set

A = {2, 4, 7, 9, 11, 12} as the domain find the range for each of the following

functions.

a) f(x) = 3x – 2

b) g(x) = 2x2 + 1

x

c) h(x) =

1 − x

Exercise 6

State the inverse of the following functions:

2

a) f (x) = 3 −

x

1

b) g(x) =

1− x

c) h(x) = 3x2 − 2

Exercise 7

Using as many different sets of real numbers as domains give examples of the

following:

a) A mapping which is not a function

b) A mapping which is a function

c) A function whose inverse is not a function

d) A function whose inverse is also a function

Demonstrate each example on a mapping diagram. If you are in a group each

member should come up with an example of his or her own for each of the cases

above.

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Module 1: Basic Mathematics

Unit 1, Activity 2: Composite functions

Specific Objectives

By the end of this activity, the learner should be able to:

Demonstrate a situation in which two consecutive instructions issued in

two different orders may yield different results.

Verify that two elementary functions operated (one after another) in two

different orders may yield different composite functions.

Draw and examine graphs of different classes of functions starting with

linear, quadratic etc.

Overview

Composite functions are about combinations of different simple mappings in order

to yield one function. The process of combining even two simple statements in

real life situations in order to yield one compound statement is important. Indeed

the order in which two consecutive instruction are issued must be seriously consi-

dered so that we do not end up with some embarrassing results.

In this activity we are set to verify that two elementary functions whose formulae

are known if combined in a certain order will yield one composite formula and

if order in which they are combined is reversed then this may yield a different

formula.

We note here that it is equally important to be able to represent a composite

function pictorially by drawing its graph and examine the shape. Indeed, the

learner will be able to draw these graphs starting with linear functions quadratic

and even trigonometric functions etc.

Key Concepts

Composite Function: This is a function obtained by combing two or more other

simple functions in a given order.

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Readings

All of the readings for the module come from Open Source text books. This

means that the authors have made them available for any reader to use them

without charge. We have provided complete copies of these texts on the CD

accompanying this course.

1. Sets relations and functions by Ivo Duntsch and Gunther Gediga, Metho-

dos publishers (UK) 2000. (File name on CD: Sets_Relations_Func-

tions_Duntsch)

Internet Resources

Composite Functions (visited 06.11.06)

http://www.bbc.co.uk/education/asguru/maths/13pure/02functions/06composite/

index.shtml

Read through the first page

Use the arrow buttons at the bottom of the page to move to the next

page

Page 2 is an interactive activity. Work through it carefully.

Read page 3 for details on notation.

Test your understanding on page 4.

Wolfram MathWorld (visited 06.11.06)

http://mathworld.wolfram.com/Composition.html

Read this entry for Composite Functions.

Follow links to explain specific concepts as you need to.

Wikipedia (visited 06.11.06)

http://www.wikipedia.org/

Type ‘Composite Functions’ into the search box and press ENTER.

Follow links to explain specific concepts as you need to.

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