week 3 relations & functions

8/3/2019 Week 3 Relations & Functions

1/25

1

Revision Exercises

Solve for the following variables

1. 2(y+3) 20

2. (3x-2)/x >5

2. 2(a+1) - 3(a+2) = 18

4. 7/x + 2 = 3/x

5. (2x+5)/3 - (x+2)/7 = 4


2/25

2

QCF157Week 3

Relations and Functions

CRICOS No 00213J


3/25

3

Set Notation

Sets are often denoted by capital lettersfor example, A = {1,2,3,4}, B = {3,4,5}

1A reads "1 is an element of set A" 1 B reads "1 is not an element set B"

A B reads "A union B" = {1,2,3,4,5}

A B reads "A intersection B" = {3,4}" A \ {3,4} reads "set A less set containing 3

&4" = {1,2}


4/25

4

Real Number System

Subsets

N = {1,2,3.} also Z+Set ofnaturalor counting numbers

W= {0,1,2,3.} also Z+U {0}Set of whole numbers

Z- = { ... -3, -2, -1 }Set of negative integers

Z = { ... -3, -2, -1, 0, 1, 2, 3, ... }Set of integers


5/25

5

Q = set of rational numbersThis set contains all quotients of integersi.e. a/b where a,bZThe decimal representation terminates or

repeats itself

e.g. 5/2= 2.5, 1/3= 0.3, I/7=0.142857142857 I = set of irrational numbersThe decimal representation does not terminate

or repeat itself.e.g. = 3.141 592 653 589 793 238 .....and 2

The combination of Q and I gives us the realnumbersR


6/25

6

Describing Sets

x

0 1 2-1-2 3-3

{x: x > -1} ( -1, )

0 1 2 3

0 1 2 3-1-2-3

-1-2-3

x {x: x 2} ( -, 2 ]

x {x: -2 x 3} [ -2, 3 ]

Examples:

Ex17.1


7/25

7

Relations and Functions

Arelation isaset of ordered pairs.eg. {(7,3),(5,-4),(9,3),(-10,2),(12,-4),(9,6)} The domain ofa relation is the set of the

first numbers of the ordered pairs.eg. the domain is {7, 5, 9, -10,12, 9} The range of a relation is the set of the

second numbers of the ordered pairs.

eg. the range is {3, -4, 3, 2, -4, 6} The values of x form the domain The values of y form the range


8/25

8

Cartesian Plane

The x-axis is the horizontal and the y-axis is the vertical

Each point has x and y coordinates, P(x,y) is theordered pair

The point P ( 1 , 2 ) is in the first quadrant The origin is where the x and y axis cross

The y-intercept is where a curve crosses the y axis

The x intercept is where a curve crosses the x axis

X axis

Y axis

First QuadrantSecond quadrant

Third quadrant Fourth quadrant


9/25

9

Representation of a Relation

1.A relation can be represented by a table ofvalues

x 7 5 9 10 12 9

y 3 -4 3 2 -4 6

2.A relation can be represented by a cartesian

plane graph

.

.

.

.

x

y

(9 , 3 )


10/25

10

Representation of a Relation

3. A relation can be represented by a mappingdiagram.

4. A relation may be represented by anequation e.g. y = x +1, y = x


11/25

11

Functions A functionis a special type of relation and can be

represented the same way. Defined as

1. No two ordered pairs in the set have the same firstelement

Is the relation {(7,3),(5,-4),(9,3)} a function?

Is the relation {(7,3),(5,-4),(9,3),(-10,2),(12,-4),(9,6)} afunction?

2. For each value of x there is only one value of y

Is the relation y = x+1 a function?

Is the relation y = x a function?3. Graphically a vertical line will cut the graph only once

see previous Cartesian plane graph


12/25

12

Examples1.Jason is in love with Jenny. He phones her several

times each day. Each time he phones her it costs25c. How can we represent the relation between thenumber of calls and the total cost?

x 0 1 2 3

y 0 0.25 0.50 0.75ii) A list of ordered pairs

{(0,0),(1,0.25),(2,0.5),(3,0.75)}

i) A table

iii) A formula C = 0.25 n

iv) A graphIs the relation a function?

What is the domain and range

.

.

.


13/25

13

Function NotationReferring to the telephone example

C = f(n) reads C is a function of nf is the name of the function0.25n is the function rule

What is the cost of 6 phone calls?

C = f(6) =0.25 6 = 1.5 ($1.50)

n is the independent variable (horizontal axis )

C is the dependent variable (vertical axis )

n is a discrete variable, ie a variable that canonly take fixed values and nothing inbetween. eg number of people, age in years


14/25

14

Function Notation A continuousvariable is a variable that can

conceivably take any real values in a given rangeeg height, weight

Example Lisa is to make a dress and the cost ofthe material is $9.50 per metre. Writing this as afunction with a function rule.

C = g(x) = 9.5x

What is the cost of material if 3.5 metres is used?

C = g(3.5) = 33.5 ($33.50)Name the dependent and independent variable?

Is the variable discrete or continuous?


15/25

15

Function Notation

Elements of the domain are values of theindependent variable.

Elements of the range are the dependentvariable.

y = f(x) y is a function of x

function named f variable named x


16/25

16

Examples

)2(

1

5

1

22

x

y

xy

xy

xy

xy

xy

}0/{},2/{

}0{,5

}0/{},0/{

2,,}0{},0{

,

RyRx

Ryx

RyRx

yRyRx

RyRx

RyRx

Find the domain and range for the following functions


17/25

17

Function Notation

Example:

f(x) = 4x 3

Find (i) f(2) (ii) f(0) (iii) f(a)

(i) f(2) = 4.(2) 3 = 5(ii) f(0) = 4.(0) 3 = -3(iii) f(a) = 4a - 3


18/25

18

Exercises

Ex 3.1

Ex 3.2


19/25

19

Continuity

A function is continuous if the curve canbe drawn without lifting pen from paper

A discontinuous curve has points that

do not exist . These are called points ofdiscontinuity

E.g. y = 1/x

y =1/(x-2)

Activity 3


20/25

20

Inverse FunctionsLet X be a set of boys

X = {a1, a2, a3, b, c1,c2}

Let Y be the set of the boys fathersY = {A, B, C}

The relation that Y is the father of X givesthe set of ordered pairs{ (a1,A) (a2,A) (a3,A) (b,B) (c1,C) (c2,C) }

and this describes a function since eachelement in the domain correspondsexactly to one element in the range.


21/25

21

Inverse Functions

The inverse of this relation, that Y is theson of X gives the set of ordered pairs

{ (A,a1) (A,a2) (A,a3) (B,b) (C,c1) (C,c2)

Note the new domain of the inverse isX = {A, B, C} which was the range of thefunction

And the range of the inverse isY = {a1, a2, a3, b, c1,c2} which was thedomain of the function


22/25

22

Inverse Functions The inverse of a function is the relation

obtained by interchanging the x and yvariables of the function

Algebraically we find the inverse of a functionby swapping x and y

eg f(x) = x + 1Let y = x + 1 become x = y + 1 which wethen express with y as the subject

y = x - 1 = f 1(x)

Note the inverse function notation can only beused if the new function is an inverse. Not allinverses will be functions.


23/25

23

Example

Find the inverse of y = 2x +2 and f(x) = x2

Are these inverses functions?

Is the inverse of y = x2, x 0 a function?


24/25

24

y = 2x + 2, x R, y R

Inverse x = 2y + 2

2y = x 2

y = x/2 1 x R, y R

f(x) = x2, x R, y 0

Inverse x = y2

y = x x 0, y R


25/25

25

Graphing Inverse Functions

What do you notice about a function andits inverse with respect to the line y = x?

y=x

Handout

week 3 relations & functions

Documents