Topic 3Topic 3

Introduction to Functions IIIntroduction to Functions II

Distinction between functions and relations Distinction between functions and relations (covered in Topic 1)(covered in Topic 1)

Distinction between continuous functions, discontinuous functions and Distinction between continuous functions, discontinuous functions and discrete functions.discrete functions.

Practical applications of quadratic functions, the reciprocal function Practical applications of quadratic functions, the reciprocal function and inverse variationand inverse variation

Relationship between the graph of Relationship between the graph of f(x)f(x) and the graphs of and the graphs of f(x)+af(x)+a, , f(x+a),f(x+a), af(x), f(ax)af(x), f(ax) for both positive and negative values of the constant a for both positive and negative values of the constant a

General shapes of graphs of absolute value functions, the reciprocal General shapes of graphs of absolute value functions, the reciprocal function and polynomial functions up to and including the fourth degreefunction and polynomial functions up to and including the fourth degree

Algebraic and graphical solution of two simultaneous equations in two Algebraic and graphical solution of two simultaneous equations in two variables (to be applied to only linear and quadratic functions variables (to be applied to only linear and quadratic functions (covered in Topic 1)(covered in Topic 1)

Concept of the inverse of a functionConcept of the inverse of a function

Composition of two functionsComposition of two functions

ContinuousContinuous functions and discontinuous functions functions and discontinuous functions

With a continuous function, you do not need to lift your pencilWith a continuous function, you do not need to lift your pencil

in order to draw the graph:in order to draw the graph:

x

y

ContinuousContinuous functions and discontinuous functions functions and discontinuous functions

With a continuous function, you do not need to lift your pencilWith a continuous function, you do not need to lift your pencil

in order to draw the graph:in order to draw the graph:

With a With a discontinuous function, you continuous function, you need toneed to lift your pencillift your pencil

in order to draw the graph:in order to draw the graph:

x

y

●

●

Discrete FunctionsDiscrete Functions

Consider the table below which shows the maximum temperature in Brisbane for the first week of July.

Date in JulyDate in July 1 2 3 4 5 6 7

Maximum Maximum Temperature (Temperature (° C)° C) 16 18 12 13 17 20 11

This is a finite sequence. Also there is no relationship between successive terms

Discrete FunctionsDiscrete FunctionsDate in JulyDate in July 1 2 3 4 5 6 7

Maximum Temperature (Maximum Temperature (° C)° C) 16 18 12 13 17 20 11

This is a finite sequence. Also there is no relationship between successive terms

Temp in July

0

5

10

15

20

25

0 1 2 3 4 5 6 7

Date

Tem

p (

Deg

C)

On the other hand, consider this sequenceOn the other hand, consider this sequence

1, 3, 5, 7, 9, …1, 3, 5, 7, 9, … It represents the odd numbers and is an It represents the odd numbers and is an

infinite series. In this sequence it is easy to infinite series. In this sequence it is easy to work out the pattern.work out the pattern.

Each number in this sequence is called a Each number in this sequence is called a termterm. Rather than using function notation, . Rather than using function notation, the terms are shown using the terms are shown using subscript subscript notationnotation..

e.g. te.g. t33 = 5 = 5

Each number in this sequence is called a Each number in this sequence is called a termterm. Rather . Rather than using function notation, the terms are shown using than using function notation, the terms are shown using subscript notationsubscript notation..

e.g. te.g. t33 = 5 = 5

The nThe nthth term of the sequence, t term of the sequence, tnn, is refered , is refered

to as the general term. In this example, the to as the general term. In this example, the general term is…general term is…

ttnn = 2n – 1 = 2n – 1

Model: For the sequence tModel: For the sequence tnn = 3n + 4 : = 3n + 4 :

(a) write down the first 4 terms(a) write down the first 4 terms(b) which term is equal to 49?(b) which term is equal to 49?

(a) t(a) tnn = 3n + 4 = 3n + 4

tt11 = 3 = 3 × 1 + 4 = 7× 1 + 4 = 7

tt22 = 3 = 3 × 2 + 4 = 10× 2 + 4 = 10

tt33 = 3 = 3 × 3 + 4 = 13× 3 + 4 = 13

tt44 = 3 = 3 × 4 + 4 = 16× 4 + 4 = 16

Model: For the sequence tModel: For the sequence tnn = 3n + 4 : = 3n + 4 :

(a) write down the first 4 terms(a) write down the first 4 terms(b) which term is equal to 49?(b) which term is equal to 49?

(b) t(b) tnn = 3n + 4 = 3n + 4

49 = 3n + 449 = 3n + 4

n = (49 – 4) n = (49 – 4) ÷ 3÷ 3

n = 15n = 15

Therefore 49 is the 15Therefore 49 is the 15thth term. term.

ExerciseExercise

Page 132 Ex 4.5 Page 132 Ex 4.5

1 – 4 1 – 4

Quadratic FunctionsQuadratic Functions

Model: Use a graph to find values of x such Model: Use a graph to find values of x such that (a) xthat (a) x22 – x – 6 = 0 – x – 6 = 0

(b) x(b) x22 + 5x + 2 = 0 + 5x + 2 = 0

How can we do this?How can we do this?

Where a graph cuts the x-axis is called a root or a

zero of the equation.

A quadratic equation A quadratic equation may have may have

2 roots2 roots

or 1 rootor 1 root

or no rootsor no roots

Model: Use a GC to draw a Model: Use a GC to draw a graph and find approximate graph and find approximate roots for the equation 4xroots for the equation 4x22 – – 3x = 103x = 10

Show how to find roots by using the “table” Show how to find roots by using the “table” on the GC and changing the increments on the GC and changing the increments to smaller and smaller sizes.to smaller and smaller sizes.

ExerciseExercise

Page 237 Ex 7.1 Page 237 Ex 7.1

11223 a,b,h,j3 a,b,h,j

Review FactorisingReview Factorising

ExerciseExercise

Page 240 Ex 7.2 Page 240 Ex 7.2 1-6 (a couple from each)1-6 (a couple from each)

Completing the SquareCompleting the Square

Solve xSolve x22 – 4x -12 = 0 by completing the square. – 4x -12 = 0 by completing the square.

xx22 – 4x -12 = 0 – 4x -12 = 0

xx22 – 4x = 12 – 4x = 12

xx22 – 4x + – 4x + (-2)(-2)22 = 12 + = 12 + 44

(x-2)(x-2)22 = 16 = 16

x-2 = x-2 = 44

x = 2 x = 2 44

x = 6 or x = -2x = 6 or x = -2

Solve xSolve x22 + 6x – 2 = 0 by completing the square. + 6x – 2 = 0 by completing the square.

xx22 + 6x = 2 + 6x = 2

xx22 + 6x + + 6x + 99 = 2 + = 2 + 99

(x+3)(x+3)22 = 11 = 11

x+3 = x+3 = 1111

x = -3x = -31111

x = .317 or x = -6.317 (3dp)x = .317 or x = -6.317 (3dp)

Solve xSolve x22 + 5x – 2 = 0 by completing the square. + 5x – 2 = 0 by completing the square.

xx22 + 5x = 2 + 5x = 2

xx22 + 5x + + 5x + 6.25 6.25 = 2 + = 2 + 6.256.25

(x+2.5)(x+2.5)22 = 8.25 = 8.25

x+2.5 = x+2.5 = 8.258.25

x = -2.5x = -2.58.258.25

x = .372 or x = -5.372 (3dp)x = .372 or x = -5.372 (3dp)

Solve 2xSolve 2x22 + 12x – 5 = 0 by completing the + 12x – 5 = 0 by completing the square.square.

2x2x22 + 12x = 5 + 12x = 5

xx22 + 6x = 2.5 + 6x = 2.5

xx22 + 6x + + 6x + 99 = 2.5 + = 2.5 + 99

(x+3)(x+3)22 = 11.5 = 11.5

x+3 = x+3 = 11.511.5

x = -3x = -311.511.5

x = .391 or x = -6.391 (3dp)x = .391 or x = -6.391 (3dp)

aacbb

aacb

ab

aacb

ab

aacb

ab

ab

aac

ab

ab

ac

ab

ab

ac

ab

x

x

x

x

x

xx

xx

cbxax

cbxaxSolve

24

24

2

44

2

442

2

4442

2

4

22

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

)(

)(

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0

Does this look familiar ?

ExerciseExercise

Page 240 Ex 7.2 Page 240 Ex 7.2 1-6 (a couple from each)1-6 (a couple from each)

7-107-10 Page 242 Ex 7.3 Page 242 Ex 7.3

3(a-g completing the 3(a-g completing the square), 4-8square), 4-8

Your GC can be used to look at a Your GC can be used to look at a familyfamily of curves: of curves:

e.g.e.g. y = xy = x22 , y = x , y = x22 + 3, y = x + 3, y = x22 – 3 – 3

e.g. y = xe.g. y = x22 , y = (x-3) , y = (x-3)22 , y = (x+3) , y = (x+3)22

Model: Find the turning point of the following curves by Model: Find the turning point of the following curves by completing the square:completing the square:(a) y = x(a) y = x22 – 4x + 7 – 4x + 7(b) y = 2x(b) y = 2x22 + 6x - 5 + 6x - 5

(a) y = x(a) y = x2 2 – 4x + 7– 4x + 7

= x= x22 – 4x – 4x ++ (-2)(-2)22 + 7 + 7 – 4– 4

= (x-2)= (x-2)22 + 3 + 3

Now y will be as small as y can be when x = 2Now y will be as small as y can be when x = 2

When x = 2, y = 3 When x = 2, y = 3

∴∴(2,3) is a turning point(2,3) is a turning point

Model: Find the turning point of the following curves by Model: Find the turning point of the following curves by completing the square:completing the square:(a) y = x(a) y = x22 – 4x + 7 – 4x + 7(b) y = 2x(b) y = 2x22 + 12x - 5 + 12x - 5

(b) y = 2x(b) y = 2x2 2 - 12x - 5- 12x - 5

= 2(x= 2(x22 - 6x) – 5 - 6x) – 5

= 2[x= 2[x22 - 3x + - 3x + (-3)(-3)22 ]] – 5 – 5 – 2 × 9– 2 × 9

= 2(x - 3)= 2(x - 3)22 – 23 – 23

Now y will be as small as y can be when x = 3Now y will be as small as y can be when x = 3

When x = 3, y = -23 When x = 3, y = -23

∴∴(3,-23) is a turning point(3,-23) is a turning point

y = -3xy = -3x22 + 12x + 7 + 12x + 7

a

b

ab

aacbb

aacbb

xt

2

222

22

42

4 22

xt

i.e. Turning points occur when x = i.e. Turning points occur when x = -b-b

2a2a

Check that this is so in previous modelCheck that this is so in previous model

ExerciseExercise

Page 246 Ex 7.4 No. 2, Page 246 Ex 7.4 No. 2, 3, 4 3, 4

Page 251 Ex 7.5 7-9 Page 251 Ex 7.5 7-9

Solving simultaneous Solving simultaneous equationsequations

Solve y = xSolve y = x22 – 4x + 6 and 2x – y = 3 – 4x + 6 and 2x – y = 3y = xy = x22 – 4x + 6 ….(1) – 4x + 6 ….(1)2x – y = 3 ….(2)2x – y = 3 ….(2)(2) (2) y = 2x – 3 ….(3) y = 2x – 3 ….(3)Equating (1) and (3) Equating (1) and (3) xx22 – 4x + 6 = 2x – 3 – 4x + 6 = 2x – 3xx22 - 6x + 9 = 0 - 6x + 9 = 0(x-3)(x-3)2 2 = 0 = 0 x = 3 x = 3 y = 3 y = 3(Check graphically)(Check graphically)

ExerciseExercise

Page 254 Ex 7.6 Page 254 Ex 7.6

Inverse VariationInverse Variation

As one quantity increases the other As one quantity increases the other decreases.decreases.

Consider 100 papers to be picked up Consider 100 papers to be picked up from off the school oval. The more from off the school oval. The more students involved, the less each picks up.students involved, the less each picks up.

No. of StudentsNo. of Students 11 22 44 55 1010 2020 2525 5050

Papers each Papers each picks uppicks up

Inverse VariationInverse Variation

As one quantity increases the other As one quantity increases the other decreases.decreases.

Consider 100 papers to be picked up Consider 100 papers to be picked up from off the school oval. The more from off the school oval. The more students involved, the less each picks up.students involved, the less each picks up.

No. of StudentsNo. of Students 11 22 44 55 1010 2020 2525 5050

Papers each Papers each picks uppicks up 100100 5050 2525 2020 1010 55 44 22

010

20304050

607080

90100

0 10 20 30 40 50

No of Students

Pap

ers

Co

llec

ted

No. of StudentsNo. of Students 11 22 44 55 1010 2020 2525 5050

Papers each Papers each picks uppicks up 100100 5050 2525 2020 1010 55 44 22

X times YX times Y 100100 100100 100100 100100 100100 100100 100100 100100

Inverse variations always fit the patternInverse variations always fit the pattern

These graphs will always have the general shape…These graphs will always have the general shape…

)(constant kyx x

ky

Our example obviously only relates to x > 0

The graph above (y=1/x) is a discontinuous graph. It has a discontinuity (does not exist) at x=0.

ExerciseExercise

Year 10 Set 12 E & FYear 10 Set 12 E & F

Model: For the function Model: For the function

(a)(a) Graph the functionGraph the function(b)(b) State the domain and rangeState the domain and range

43

2

t

y

There will be a discontinuity when t-3=0

t = 3 is an asymptote

Horizontal asymptote at y = -4

As t ↑ 3, h(t) → ∞

As t ↓ 3, h(t) → -∞

When t = 0, h(t) = -3⅓

When h(t) = 0, t = 2½

-3⅓

2½

Domain: all real numbers except 3

Range: all real numbers except -4

Attempt to graph 3)2(

12

x

y

Domain: any real, except -2 Range: y > 3

ExerciseExercise

Photocopied sheets of Photocopied sheets of ““Graphing a FunctionGraphing a Function””

Inverse of a FunctionInverse of a Function

Take a function and swap the x and y Take a function and swap the x and y pronumerals. e.g.pronumerals. e.g.

y = 3x + 2y = 3x + 2 becomesbecomes x = 3y + 2x = 3y + 2rearrangerearrange (x – 2)/3 = y(x – 2)/3 = y

y = y = ⅓x - ⅔⅓x - ⅔

Graph both functionsGraph both functions

These lines are a reflection in the line y = x

ExerciseExercise

Photocopied sheets of Photocopied sheets of ““Inverse of a FunctionInverse of a Function””

Composition of Two FunctionsComposition of Two Functions

ConsiderConsider

f(x) = xf(x) = x22 – 3x + 2 – 3x + 2

f(3)f(3) = 3= 322 – 3 × 3 + 2 – 3 × 3 + 2

= 2= 2

f(a) f(a) = a= a22 – 3a + 2 – 3a + 2

If g(x) = 2x + 1If g(x) = 2x + 1

f [g(x)] f [g(x)]

= f( = f( 2x + 12x + 1))

= = ((2x+12x+1))22 – 3( – 3(2x+12x+1) + 2) + 2

= 4x= 4x22 + 4x + 1 – 6x – 3 + 2 + 4x + 1 – 6x – 3 + 2

= 4x= 4x22 – 2x – 2x

Try These … Try These …

1.1.

f(x) = xf(x) = x22 + 5x – 2 + 5x – 2

g(x) = 3x – 2 g(x) = 3x – 2

FindFind

1.1. f [g(x)]f [g(x)]

2.2. g [f(x)]g [f(x)]

3.3. g [g(x)]g [g(x)]

2.2.

f(x) = 2xf(x) = 2x22 + 5x – 4 + 5x – 4

g(x) = 5 – 3x g(x) = 5 – 3x

FindFind

1.1. f [g(x)]f [g(x)]

2.2. g [f(x)]g [f(x)]

3.3. g [g(x)]g [g(x)]

Try These … Try These …

1.1.

f(x) = xf(x) = x22 + 5x – 2 + 5x – 2

g(x) = 3x – 2 g(x) = 3x – 2

FindFind

1.1. f [g(x)] f [g(x)] = 9x= 9x22 + 3x – 8 + 3x – 8

2.2. g [f(x)] g [f(x)] = 3x= 3x22 + 15x – 8 + 15x – 8

3.3. g [g(x)] g [g(x)] = 9x – 8 = 9x – 8

2.2.

f(x) = 2xf(x) = 2x22 + 5x – 4 + 5x – 4

g(x) = 5 – 3x g(x) = 5 – 3x

FindFind

1.1. f [g(x)]f [g(x)] = 18x = 18x22 – 75x + 71 – 75x + 71

2.2. g [f(x)]g [f(x)] = 17 – 15x – 6x = 17 – 15x – 6x22

3.3. g [g(x)]g [g(x)] = 9x – 10 = 9x – 10

When functions are contained within other functions, they can be simplified as follows.

y = (x2 - 5)2 - 5(x2 - 5) + 6

x2 - 5 is a function, let u = x2 - 5

y = u2 - 5u + 6

When functions are contained within other functions, they can be simplified as follows.

y = (x2 - 5)2 - 5(x2 - 5) + 6

x2 - 5 is a function, let u = x2 - 5

y = u2 - 5u + 6

solve (x2 - 5)2 - 5(x2 - 5) + 6 = 0

Function of a Function handout sheetsFunction of a Function handout sheets