- M208_4Pure mathematics
Real functions and graphs
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This free course is an adapted extract from the Open Unviersity course M208: *Pure Mathematics* www3.open.ac.uk/study/undergraduate/course/m208.htmThis version of the content may include video, images and interactive content that may not be optimised for your device. You can experience this free course as it was originally designed on OpenLearn, the home of free learning from The Open University: www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/real-functions-and-graphs/content-section-0.There you’ll also be able to track your progress via your activity record, which you can use to demonstrate your learning.The Open University, Walton Hall, Milton Keynes, MK7 6AACopyright © 2016 The Open University
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978-1-4730-0613-3 (.epub)IntroductionMany 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*After studying this course, you should be able to:understand the definition of a real functionuse the notation for intervals of the real linerecognise and use the graphs of the basic functions described in the audio sectionunderstand the effect on a graph of translations, scalings, rotations and reflectionsunderstand how the shape of a graph of a function features properties of the function such as increasing, decreasing, even and odd.1 OverviewA 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 2*x*^{2} − 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.
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.2 Real functionsIn 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 1This 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).Frames 1-4Click play to listen to the audio clip for frames 5 to 9 (7 minutes).Frames 5-9Click play to listen to the audio clip for frames 10 to 13 (7 minutes).Frames 10-13Click play to listen to the audio clip for frames 14 to 17 (7 minutes).Frames 14-17Click play to listen to the audio clip for frames 18 to 21 (3 minutes).Frames 18-21Click play to listen to the audio clip for frames 22 to 25 (5 minutes).Frames 22-253 Graph sketchingIn 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 24 New graphs from oldIn 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 35 Hyperbolic functionsIn 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 46 Curves from parametersIn 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 57 Solutions to the exercisesSection 6 contains solutions to the exercises that appear throughout sections 1-5.Click the link below to open the solutions (13 pages, 232KB).Section 6ConclusionThis 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.Keep on learning Study another free courseThere are more than **800 courses on OpenLearn** for you to choose from on a range of subjects. Find out more about all our free courses. Take your studies furtherFind out more about studying with The Open University by visiting our online prospectus. If you are new to university study, you may be interested in our Access Courses or Certificates. What’s new from OpenLearn?
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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.**Don't miss out:**If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University - www.open.edu/openlearn/free-courses