Real functions and graphs
M208_4 Pure mathematics
Real functions and graphs
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Real functions and graphs
About this free course
This free course is an adapted extract from the Open Unviersity course M208: Pure
Mathematics www3.open.ac.uk/study/undergraduate/course/m208.htm
This version of the content may include video, images and interactive content that
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There you’ll also be able to track your progress via your activity record, which you
can use to demonstrate your learning.
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Real functions and graphs
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978-1-4730-1381-0 (.kdl)
978-1-4730-0613-3 (.epub)
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Contents

Introduction

Learning outcomes

1 Overview

2 Real functions

3 Graph sketching

4 New graphs from old

5 Hyperbolic functions

6 Curves from parameters

7 Solutions to the exercises

Conclusion

Keep on learning

Acknowledgements
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Real functions and graphs
Introduction
Many problems are best studied by working with real functions, and the properties of
real functions are often revealed most clearly by their graphs. Learning to sketch such
graphs is therefore a useful skill, even though computer packages can now perform
the task. Computers can plot many more points than can be plotted by hand, but
simply ‘joining up the dots’ can sometimes give a misleading picture, so an
understanding of how such graphs may be obtained remains important. The object of
this course is to review the various techniques for sketching graphs that you may have
met in your previous studies, and to extend these methods.
This OpenLearn course is an adapted extract from the Open Unviersity course M208:
Pure Mathematics
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Learning outcomes
After studying this course, you should be able to:



understand the definition of a real function
use the notation for intervals of the real line
recognise and use the graphs of the basic functions described in the
audio section
 understand the effect on a graph of translations, scalings, rotations and
reflections
 understand how the shape of a graph of a function features properties
of the function such as increasing, decreasing, even and odd.
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1 Overview
A fundamental concept in mathematics is that of a function.
Consider, for example, the function f defined by
This is an example of a real function, because it associates with a given real number x
the real number 2x2 − 1: it maps real numbers to real numbers.
One way of picturing this function is the following. First, draw up a table of values,
listing in the first row several values of x, and in the second row the corresponding
values of f(x); for example, f(0.8) = 2(0.8)2 − 1 = 0.28.
Each column in the table is essentially an ordered pair of the form (x, f(x)), which can
be plotted as a unique point in the plane. We refer to the set of all such points as the
graph of f.
Of course, it is not possible to plot all the points of the graph for a function like f,
which is defined for infinitely many values of x. Fortunately, for many graphs plotting
a few points provides a good idea of what the graph looks like. For any so-called
‘well-behaved’ function, these points can be joined up by a smooth curve – and
extended if necessary – to complete the picture.
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No doubt you are already familiar with this method of drawing a graph. However, it is
not always the most efficient method to use when knowledge about the key features of
the graph is all that is wanted. For the purposes of this course, that is all that is usually
required.
Please note that this course is presented through a series of downloadable PDF
documents.
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2 Real functions
In Section 1 we formally define real functions and describe how they may arise when
we try to solve equations. We remind you of some basic real functions and their
graphs, and describe how some of the properties of these functions are featured in
their graphs.
Click the link below to open Section 1 (12 pages, 1.8MB).
Section 1
This PDF contains an audio section in which we revise the properties of some familiar
graphs, and introduce some concepts which form the basis of the graph-sketching
strategy given in Section 2. Please listen to the audio clips below when you are
instructed to do so. Do not worry if you do not fully understand all the details at this
stage, but concentrate on the main ideas.
Click play to listen to the audio clip for frames 1 to 4 (6 minutes).
Audio content is not available in this format.
Frames 1-4
Click play to listen to the audio clip for frames 5 to 9 (7 minutes).
Audio content is not available in this format.
Frames 5-9
Click play to listen to the audio clip for frames 10 to 13 (7 minutes).
Audio content is not available in this format.
Frames 10-13
Click play to listen to the audio clip for frames 14 to 17 (7 minutes).
Audio content is not available in this format.
Frames 14-17
Click play to listen to the audio clip for frames 18 to 21 (3 minutes).
Audio content is not available in this format.
Frames 18-21
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Click play to listen to the audio clip for frames 22 to 25 (5 minutes).
Audio content is not available in this format.
Frames 22-25
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3 Graph sketching
In Section 2 we describe how the graphs of polynomial and rational functions may be
sketched by analysing their behaviour – for example, by using techniques of calculus.
We assume that you are familiar with basic calculus and that its use is valid. In
particular, we assume that the graphs of the functions under consideration consist of
smooth curves.
Click the link below to open Section 2 (16 pages, 200KB).
Section 2
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4 New graphs from old
In Section 3 we consider how to sketch the graphs of more complicated functions,
sometimes involving trigonometric functions. We look at graphs which are sums,
quotients and composites of different functions, and at those which are defined by a
different rule for different values of x.
Click the link below to open Section 3 (7 pages, 133KB).
Section 3
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5 Hyperbolic functions
In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are
constructed from exponential functions. These hyperbolic functions share some of the
properties of the trigonometric functions but, as you will see, their graphs are very
different.
Click the link below to open Section 4 (5 pages, 104KB).
Section 4
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6 Curves from parameters
In Section 5 we show how functions may be used to sketch curves in the plane, even
when these curves are not necessarily the graphs of functions.
Click the link below to open Section 5 (8 pages, 151KB).
Section 5
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7 Solutions to the exercises
Section 6 contains solutions to the exercises that appear throughout sections 1-5.
Click the link below to open the solutions (13 pages, 232KB).
Section 6
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Conclusion
This free course provided an introduction to studying Mathematics. It took you
through a series of exercises designed to develop your approach to study and learning
at a distance and helped to improve your confidence as an independent learner.
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Real functions and graphs
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Real functions and graphs
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Acknowledgements
These extracts are from M208 © 2006 The Open University.
All material contained within this course originated at The Open University.
Course image: Matt in Flickr made available under Creative Commons Attribution 2.0
Licence.
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