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[PDF] Functions I - Australian Mathematical Sciences Institute

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A guide for teachers - Years 11 and 12

1 2 3 4 5 6 7 8 9 1 0 1 1 1

2Supporting Australian Mathematics ProjectFunctions: Module 5

Functions I

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Functions I

A guide for teachers (Years 11-12)

Principal author: Associate Professor David Hunt, University of NSW

Peter Brown, University of NSW

Dr Michael Evans, AMSI

Associate Professor David Hunt, University of NSW

Dr Daniel Mathews, Monash University

Assumed knowledge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 The concept of a function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Domains and ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 Function notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 Finding domains and ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 Links forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 Functions between sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 Functions between finite sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 More examples of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 Answers to exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

Functions I

Assumed knowledge

•Familiarity with elementary set theory, as discussed in the TIMES moduleSets and

Venn diagrams(Years 7-8).

•Familiarity with the algebraic techniques described in the TIMES moduleFormulas (Years 9-10). •The content of the modulesAlgebra reviewandCoordinate geometry. •Basic knowledge of the modulesTrigonometric functions and circular measureand

Exponential and logarithmic functions.

Motivation

The expressionyAEx2, which links the two pronumeralsxandy, can be thought of in several different ways. equation. These ordered pairs determine agraph. In this case, the graph is the standard parabola.y x 0 y x

A guide for teachers - Years 11 and 12

•{5} Aformulais an equation relating different quantities using algebra. SoyAEx2is also a formula. In fact,yAEx2is an example of afunction, in the sense that each value ofx uniquely determines a value ofy. In this module, we will study the concept of a function. The formulaAAE¼r2givesA as a function ofr. The formulaVAE¼r2hexpressesVas a function of the two variables randh. In this module, we will only consider functions of one variable, such as the polynomial yAE(x¡1)(x¡2)(x¡3)(x¡4). A clear understanding of the concept of a function and a familiarity with function nota- tion are important for the study of calculus. The use of functions and function notation in calculus can be seen in the moduleIntroduction to differential calculus.

Content

Set theory

It is not possible to discuss functions sensibly without using the language and ideas of elementary set theory. In particular, we will use the following ideas from the moduleSets and Venn diagrams(Years 7-8): •sets and their elements •equal sets •listing the members of a set; for example,AAE{2,4,6,8} •set membership; for example, 42Aand 5ÝA •finite and infinite sets, and the number of elements in a finite set •subsets •unions and intersections. Sets must be well defined, and if sets are defined using mathematical notation there are rarely any problems. A set iswell definedif it is possible to determine whether or not a given object belongs to that set or not. The set of all words in the English language is not well defined.Exercise 1 Think of several reasons why 'the set of all English words" is not well defined. {6}•Functions I

Further notation

Set-builder notation is useful for describing subsets of the real numbers. For example, consider the set defined by

SAE{x2Rj¡5·x·5}

or, equivalently,

SAE{x2R:¡5·x·5}.

This can be read as 'Sis the set of allxbelonging toRsuch that¡5·x·5". In this construction: •Sis the name of the set •{ } holds the definition together •x2Rsays thatxis a real number •bothjand : mean 'with the property that" •¡5·x·5 limits the allowed values ofx. SoSis the following interval:55If we define the setTAE{x2R:jxj·5}, thenSAET. Union and intersection are familiar operations on sets. Another useful operation on sets isset difference. For two setsAandB, we define

A\BAE{x2AjxÝB}.

That is,A\Bis the set of all elements ofAthat are not inB. We can readA\Bas 'Atake awayB" or 'AminusB". For example, the set of all non-zero real numbers can be written as {x2Rjx6AE0} or, more simply, asR\{0}.

TheCartesian productof two setsAandBis defined by

A£BAE{(a,b)ja2Aandb2B}.

That is,A£Bis the set of all ordered pairs (a,b) withainAandbinB. So, for example,

R£RAE{(x,y)jx2Randy2R}.

We writeR£RasR2, which explains the use of the notationR2to describe the coordinate plane. This was introduced in the moduleCoordinate geometry.

A guide for teachers - Years 11 and 12

•{7}

Note. In this module and many others:

•Rdenotes the set of real numbers •Qdenotes the set of rational numbers •Zdenotes the set of integers •Cdenotes the set of complex numbers.

The concept of a function

When a quantityyisuniquely determinedby some other quantityxas a result of some rule or formula, then we say thatyisafunctionofx. (In other words, for each value ofx, there isat most onecorresponding value ofy.) We begin with six examples in which bothxandyare real numbers: •yAE2x•yAE1x •yAElog2x. We draw their graphs in the usual way, with thex-axis horizontal and they-axis vertical.y x

0y = x + 2

0001?2

1 1yy x xy = 3x ? 7 0 7

1y = sin x

360360

y xyy xxy = 2 x 1 x

y =y = log xThe first four functions are similar in that their formulas 'work" for all real numbersx.

ForyAE1x

, we clearly needx6AE0, and foryAElog2x, we needxÈ0. We will discuss this further in the sectionDomains and ranges. {8}•Functions I

What is a relation?

There are many naturally occurring formulas whose graphs are not the graphs of func- tions. For example:y x

0x? + y? = 25

5 5 5 5 y x 0 3 3 4 4 x 2 16 + = 1 y 2 9 y x 0 y = x y x 0 44
x 2 16 = 1 y 2

9The first graph is a circle, the second is an ellipse, the third is two straight lines, and the

fourth is a hyperbola. In each example, there are values ofxfor which there are two values ofy. So these are not graphs of functions. It turns out that the most useful concept to help describe and understand this issue is very general.

Definition

Arelationon the real numbers is any subset ofR£R. That is, a relation on the reals is a set of ordered pairs of real numbers. Thus the four graphs above and the graphs of the six example functions are all relations on the real numbers. Indeed, the graph of any function is a relation. Formally speaking, afunctionis a relation such that, for eachx, there is at most one ordered pair (x,y).

A guide for teachers - Years 11 and 12

•{9}Example The lineyAExdivides the number plane into three relations. •The relationR1AE{(x,y)jxAEy} is the line itself. Note that, for eachx, there is only one value ofy.y x

0x = y•The relationR2AE{(x,y)jxÇy} consists of all points strictly above the lineyAEx.

Note that, for eachx, there are infinitely many values ofy. y x

0x < y•The relationR3AE{(x,y)jxÈy} consists of all points strictly below the lineyAEx.

y x 0 x y {10}•Functions I We can generalise the previous example to any line in the plane, as follows.Example The equation of a linelin the plane is given byaxÅbyÅcAE0. This line determines, in a natural way, three relations on the reals: R

1AE{(x,y)jaxÅbyÅcAE0}

2AE{(x,y)jaxÅbyÅcÇ0}

3AE{(x,y)jaxÅbyÅcÈ0}.Example

Consider the circlex2Åy2AEr2, for somerÈ0. There are three relations closely con- nected with this circle. •The circle itself: R

1AE{(x,y)jx2Åy2AEr2}.

•The interior of the circle: R

2AE{(x,y)jx2Åy2Çr2}.

•The exterior of the circle: R

3AE{(x,y)jx2Åy2Èr2}.Graphs and the vertical-line test

We have seen the graphs of several naturally described functions. A sensible question to ask, for a given graph inR2, is whether it is the graph of a function. Thevertical-line testgives a simple geometric test for answering this question: If we can draw a vertical linexAEcthat cuts the graph more than once, then the graph is not the graph of a function.

A guide for teachers - Years 11 and 12

•{11} Returning to the graph ofx2Åy2AE25, we see that the vertical linexAE3 meets the graph at both (3,4) and (3,¡4).y x

03x? + y? = 25

5 5 (3,

4)5(3,4)

5Hence, the graph ofx2Åy2AE25 is not the graph of a function. The linexAE6 does not

meet the graph at all, but this does not matter. In general: •for¡5ÇcÇ5, the linexAEcmeets the graph twice •forcAE¡5 and forcAE5, the linexAEcmeets the graph once •forcÇ¡5 and forcÈ5, the linexAEcdoes not meet the graph.

Relations which determine functions

quotesdbs_dbs4.pdfusesText_8

[PDF] [PDF] Functions I - Australian Mathematical Sciences Institute

A guide for teachers - Years 11 and 12

2Supporting Australian Mathematics ProjectFunctions: Module 5

Functions I

Published by Education Services Australia

PO Box 177

Carlton South Vic 3053

Australia

Tel: (03) 9207 9600

Fax: (03) 9910 9800

Email: info@esa.edu.au

Website: www.esa.edu.au

Employment and Workplace Relations.

Supporting Australian Mathematics Project

Australian Mathematical Sciences Institute

Building 161

The University of Melbourne

VIC 3010

Email: enquiries@amsi.org.au

Functions I

A guide for teachers (Years 11-12)

Peter Brown, University of NSW

Dr Michael Evans, AMSI

Associate Professor David Hunt, University of NSW

Dr Daniel Mathews, Monash University

Functions I

Assumed knowledge

Venn diagrams(Years 7-8).

Exponential and logarithmic functions.

Motivation

A guide for teachers - Years 11 and 12

Content

Set theory

Further notation

SAE{x2Rj¡5·x·5}

SAE{x2R:¡5·x·5}.

A\BAE{x2AjxÝB}.

TheCartesian productof two setsAandBis defined by

A£BAE{(a,b)ja2Aandb2B}.

R£RAE{(x,y)jx2Randy2R}.

A guide for teachers - Years 11 and 12

Note. In this module and many others:

The concept of a function

0y = x + 2

0001?2

1y = sin x

360360

ForyAE1x

What is a relation?

0x? + y? = 25

9The first graph is a circle, the second is an ellipse, the third is two straight lines, and the

Definition

A guide for teachers - Years 11 and 12

0x = y•The relationR2AE{(x,y)jxÇy} consists of all points strictly above the lineyAEx.

0x < y•The relationR3AE{(x,y)jxÈy} consists of all points strictly below the lineyAEx.

1AE{(x,y)jaxÅbyÅcAE0}

2AE{(x,y)jaxÅbyÅcÇ0}

3AE{(x,y)jaxÅbyÅcÈ0}.Example

1AE{(x,y)jx2Åy2AEr2}.

2AE{(x,y)jx2Åy2Çr2}.

3AE{(x,y)jx2Åy2Èr2}.Graphs and the vertical-line test

A guide for teachers - Years 11 and 12

03x? + y? = 25

4)5(3,4)

5Hence, the graph ofx2Åy2AE25 is not the graph of a function. The linexAE6 does not

Relations which determine functions