Unit 3.pdf - Unit 3 Functions Introduction Introduction In...

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Unit 3Functions
IntroductionIntroductionIn mathematics you often work with situations in which one quantityA Ferris wheeldepends on another. For example:The distance walked by a woman at a particular speed depends on thetime that she’s been walking.The height of a gondola on a Ferris wheel depends on the anglethrough which the wheel has rotated since the gondola was in itslowest position.The number of 5-litre tins of a particular type of paint needed by adecorator depends on the area that he intends to paint.Whenever one quantity depends on another, we say that the first quantityis afunctionof the second quantity. The idea of a function isfundamental in mathematics, and in particular it forms the foundation forcalculus, which you’ll begin to study in Unit 6.In this unit you’ll be introduced to the terminology and notation that areused for functions. You’ll learn about some standard, frequently-arisingtypes of functions, and how to use graphs to visualise properties offunctions. You’ll also learn how you can use your knowledge about a fewstandard functions to help you understand and work with a wide range ofrelated functions. Later in the unit you’ll reviseexponential functionsandlogarithms, and practise working with them. In the final section you’llreviseinequalities, and see how working with functions and their graphscan help you understand and solve some quite complicated inequalities.This is a long unit. The study calendar allows extra time for you tostudy it.1 Functions and their graphsThis section introduces you to the idea of a function and its graph, andshows you some standard functions. You’ll start by learning aboutsets,which are needed when you work with functions and also in many otherareas of mathematics.201
Unit 3Functions1.1Sets of real numbersIn mathematics asetis a collection of objects. The objects could beanything at all: they could be numbers, points in the plane, equations oranything else. For example, each of the following collections of objectsforms a set:all the prime numbers less than 100all the points on any particular line in the planeall the equations that represent vertical linesthe solutions of any particular quadratic equation.A set can contain any number of objects. It could contain one object, twoobjects, twenty objects, infinitely many objects, or even no objects at all.Each object in a set is called anelementormemberof the set, and wesay that the elements of the setbelong toor areinthe set.There are many ways to specify a set. If there are just a few elements,then you can list them, enclosing them in curly brackets. For example, youcan specify a setSas follows:S={3,7,9,42}.Another simple way to specify a set is to describe it. For example, you cansay ‘letTbe the set of all even integers’ or ‘letUbe the set of all realnumbers greater than 5’. We usually denote sets by capital letters.The set that contains no elements at all is called theempty set, and isdenoted by the symbol.

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Complex number, John Venn

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