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  • Active Maths 1

    Michael Keating, Derek Mulvany and James OLoughlinSpecial Advisors: Oliver Murphy, Colin Townsend and Jim McElroy

    2r

    Junior certificate

    Project Maths Strand 5

    y 2 -

    y 1x 2

    - x

    1

    m =

    r2

  • Editor: Priscilla OConnor, Sarah Reece

    Designer: Liz White

    Layout: Compuscript

    Illustrations: Compuscript, Denis M. Baker, Rory ONeill

    ISBN: 978-1-78090-165-71383

    Michael Keating, Derek Mulvany, James OLoughlin and Colin Townsend, 2012

    Folens Publishers, Hibernian Industrial Estate, Greenhills Road, Tallaght, Dublin 24, Ireland

    Acknowledgements

    The authors would like especially to thank Jim McElroy for his work on the written solutions and his invaluable advice.

    The authors and publisher wish to thank Thinkstock for permission to reproduce photographs.

    All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from the publisher. The publisher reserves the right to change, without notice, at any time, the specification of this product, whether by change of materials, colours, bindings, format, text revision or any other characteristic.

  • Strand 5 Chapters

    Chapter 30 Functions ...............................................498 30.1 Functions ...............................................499 Idea of a Function ..................................499 Naming Functions ..................................500 Mapping Diagrams .................................500 Couples ..................................................500 InputOutput Table ................................501 Domain ..................................................501 Range .....................................................501 Codomain ..............................................501 30.2 Function Notation .................................505 Codomain ..............................................506

    Chapter 31 Graphing Functions ..............................511 31.1 Linear Functions ....................................512 31.2 Quadratic Functions .............................515 31.3 Applications of Graphs .........................516 Solve f(x) = 0 .........................................516 Solve f(x) = q, Where q Is a Number ......517 Find the Value of f(p) ..............................517 Solve f(x) = g(x) ......................................518 31.4 Transformation of Linear Functions ......523 Linear Functions .....................................523 31.5 Transformations of Quadratic

    Functions ...............................................527 Graphs of the Form y = ax2 ....................527 Graphs of the Form y = x2 + b ...............528 Graphs of the Form y = (x + b)2 .............529

    Contents

    Strand 5 Activities

    Chapter 30 Functions ...............................................203

    Chapter 31 Graphing Functions ..............................205

    Answers

  • Strand 5 Chapters

  • 30ch

    apte

    rch

    apte

    r

    Engage with the concept of a function, domain, codomain and range

    Make use of function notation f (x) = f: x f: x y =

    Learning OutcomesIn this chapter you will learn to:

    Functions

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    30.1 FunctionsIn mathematics, a relation is a rule that describes how certain things are linked to each other. For example, the harder you kick a ball, the further it will travel.

    There are many examples of relations in real life where one thing is related to another.

    Your fitness is related to how much exercise you do.

    Your arm length is related to your height.

    The area of a circle is related to its radius.

    idea of a FunctionA function is a special type of relation. When thinking about functions, it can be easier to think about them as a type of machine or calculator.

    In a function, the number you put in (the input) is changed into another number (the output) according to a certain relationship (the rule).

    3

    6

    Rule: Double input

    (Input)

    (Output)

    In the above function, each input is doubled.

    In a function, each input gives back one output only. For example, an input of 4 into the function machine shown will only ever give an answer of 8 (2 4). It can never give back any other answer.

    A function does allow for two inputs to give the same output.

    2 (Input)

    (Output) 4

    Rule: (input)2

    2 (Input)

    (Output) 4

    Rule: (input)2

    In the function machine shown, every input is squared. Inputs of 2 and 2 will both have the same output, 4.

    KEY W ORDS

    YOU SHOULD REMEMBER.. .

    function input

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    For an everyday example of functions, consider the barcodes used to scan goods in shops.

    In a shop, if a barcode is scanned (inputted) at a checkout (function), then only one price (output) will appear. The register will not show two prices for the same barcode.

    Input: Barcode Output: Shows that the item costs 1

    However, when the checkout shows that the price of an item is 1, there could be many goods in the shop that cost 1.

    Output: 1 Input: Numerous items in the shop could cost 1.

    naming FunctionsIt is useful to give a function a name. The most common name is f, but other names can also be used, such as g or h (or any other name we want).

    Mapping DiagramsOne method of showing a function is to draw a mapping diagram. In a mapping diagram, each input of the function is mapped to its output.

    The mapping diagram below shows the function g. In the function g, each input is multiplied by 3 and then 1 is added.

    An input of 4 will result in an output of 13. So 4 is mapped onto 13. This is illustrated by the arrows in the diagram.

    When a function is shown using a mapping diagram, no input will map onto two or more outputs.

    However, a function does allow for two inputs to be mapped to the same output.

    Input Output

    f is not a function

    f

    f is a function

    Input Outputf

    couplesA function can also be written as a set of couples or ordered pairs. In each couple, the first element (input) is related to the second element (output).

    Each couple or ordered pair is written as (Input, Output).

    f = {(a,1), (b,4), (c,3), (d,5)} are the couples or ordered pairs of the function f shown on the right.

    When a function is written as a set of couples, no two couples will have the same input.

    The couples: {(1,3), (3,5), (2,4)} represent a function, as no two couples have the same input.

    The couples: {(1,3), (1,2), (3,4)} do not represent a function, as the input 1 appears twice.

    ab

    c

    f

    d

    1

    4

    3

    5

    4

    Input Outputg

    1

    0

    13

    4

    1

    Rule: Multiply by 3and add 1

    Rule: Multiply by 3 and add 1

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    inputoutput tableAn inputoutput table can be used to show functions.

    Input Rule: Double the input and add 4 Output1 2(1) + 4 63 2(3) + 4 104 2(4) + 4 12

    We can then write this information as a set of couples: {(1,6), (3,10), (4,12)}.

    Consider the function h as shown.

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    h

    36789

    DomainThe domain is the set of values that you can put into a function. It is the set of all inputs. It is the set of all the first elements of the ordered pairs.

    In the diagram above, {1, 4, 5} is the domain of the function h.

    RangeThe range is the set of values that you get out of a function when you input the domain values. It is the set of all actual outputs. It is the set of all the second elements of the ordered pairs.

    In the diagram above, {3, 6, 7} is the range of the function h.

    codomainIn the diagram above, {3, 6, 7, 8, 9} is the codomain of the function h. It is the set of all elements into which the function maps.

    ACTIVITY 30.1

    range

    image

    Worked Example 30.1

    A mapping diagram of the relation R is shown.

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    1

    2

    3

    R

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    (i) Identify the domain.

    (ii) Identify the range.

    (iii) Identify the codomain.

    (iv) List the ordered pairs.

    (v) Is the relation R a function? Explain your answer.

    Solution (i) Domain = {1, 0, 1}

    (ii) Range = {2, 3, 4, 5}

    (iii) Codomain = {2, 3, 4, 5, 6}

    (iv) Ordered pairs: {(1,2), (0,3), (1,4), (1,5)}

    (v) The relation R is not a function because the input 1 has two outputs: 4 and 5.

    codomain

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    Worked Example 30.2

    (i) From the following ordered pairs, draw a mapping diagram to illustrate the relation T.

    T = {(5,25), (1,1), (2,4), (6,36), (5,25)}

    (ii) Is the relation T a function?Explain your answer.

    Solution (i) RangeTDomain

    5

    5

    1

    2

    6

    25

    1

    4

    36

    (ii) T is a function. Each input has only one output.

    Worked Example 30.3

    A function machine is shown.

    Find the range of this function and write the function as a set of ordered pairs.

    SolutionIt may be easier to use an inputoutput table to show this function.

    Input Rule: Square the input and subtract 1

    Output

    1 (1)2 1 0

    3 (3)2 1 8

    2 (2)2 1 3

    4 (4)2 1 15

    0 (0)2 1 1

    Range: {0, 8, 3, 15, 1}

    Ordered pairs: {(1,0), (3,8), (2,3), (4,15), (0,1)}

    {1, 3, 2, 4, 0}

    Rule: Square inputan