Muqaddimah

Antara keajaiban solat yang ditemui :

1. Umat Islam bertuah kerana solat wajib yang dilakukan 5 waktu sehari menyamai

aktiviti senaman dan regangan yang memperkuatkan sistem otak dan fizikal.

2. Komposisi tubuh terbaik bagi seorang manusia boleh diperolehi dengan solat

wajib lima kali sehari, kefahaman terhadap apa yang dibaca dalam solat, solat

berjamaah, dan rukuk dan sujud dengan tepat/betul.

3. Didapati orang yang solat berjamaah lebih tenang dan sihat berbanding orang

yang solat bersendirian.

4. Solat boleh merawat ED (erectly disfunction) atau lebih dikenali sebagai mati

pucuk.

5. Menurut pakar, pesakit jantung perlu melakukan senaman 30 minit setiap hari

dan ini menyamai solat wajib yang lima. 1

Functions and Graphs Constant Functions kxf )( , k is a constant and Rk . RD f and kR f

Example a) 2)( xf RD f and 2fR

b) 3)( xf RD f and 3fR

y

x

2

y

x

3

2

Linear Function General form is baxxf )( , Rba , and 0a . RD f and RR f

Example 23)( xxf intercept ,2)0( yf

xxxf ,32,0)( interept

The graph of a linear function can simply be sketched using two points usually intercepts of the axes, and then joining these two points using straight line. The graph of a linear function can also be obtained using infinite sets of elementary translations of its basic function.

y

x

2

32

3

Example How to find the graph of 23)( xxg using the graph of xxf 3)( . Note : For a linear function )(xf , )()( xfxf , we say reflect about one axis. Solution i) 2)(2323)( xfxxxg xxf 3)( is reflected about one axis and shifted upward 2 units. ii) 5)1(23513)( xfxxxg xxf 3)( is reflected about one axis and shifted to the left 1 unit and then upward 5 units. iii) 1)1(23113)( xfxxxg xxf 3)( is reflected about one axis and shifted to the right 1 unit and then downward 1 unit.

, ,

4

iv) 8)2(23823)( xfxxxg xxf 3)( is reflected about one axis, shifted to the left 2 units and then shifted upward 8 units. v) 4)2(23423)( xfxxxg xxf 3)( is reflected about one axis and shifted to the right 2 units and then downward 4 units.

vi) 21

2123

21

213)( xfxxxg

xxf 3)( is reflected about one axis and shifted to the right 21 unit and then upward

21 unit.

etc. Note : Since )()( xfxf for a linear function )(xf , we simply say reflect about one axis.

, ,

5

Quadratic Funtions General form is

.0 and , ,,)( 2 aRcbacbxaxxf

The basic form is .0 and ,)( 2 aRaaxxf

RD f and ,0fR .

Example Sketch the graphs of 2)( axxf for

41 and

21,3,2,1 aaaaa .

What can you say about the domain and range of the graphs?

y

x

y

x

6

Example How to obtain the graph of the following functions using the graph of the basic function 22)( xxf .

State the domain and range of each function.

a) 122)( 2 xxg b) 642)( 2 xxxh Solution RD f and ,0fR .

a) )1())2((1)2(122)( 2 xfxfxxg Shift the graph to the left 2 units and then downward 1 unit. RDg and ,1gR .

b) 6)2(2642)( 22 xxxxxh

6)112(2 2 xx

4)1(4)1(2 2 xfx . Shift the graph to the right 1 unit and then upward 4 units. RDh and ,4hR .

7

Example Sketch the graph each of the following functions by indicating the intercepts if any. State the domain and range of each function.

a) 223)( xxxg b) 222)( 2 xxxh

Solution a) 3)2(3223)( 222 xxxxxxxg

4)1(3)112( 22 xxx

2)(;4)1( xxfxf

intercept 3)0( yg intercepts 1,30)( xxxg

RDg and 4,gR .

y

x

8

Solution b) 2)(2222)( 22 xxxxxh

23

2122

21

212

2222

xxx

22)(;2

3

2

1xxfxf

intercept 2)0( yh

intercept no hasimpossible23

2120)(

2

xxxh

RDh and 23,hR .

y

x

9

Square root Function

The basic functuin is 0,)( axaxf and the graph is shown below.

,0fD and ,0fR .

Example Sketch thegraphs of xaxf )( for 21 and 3,2,1 aaaa .. Discuss their domain and range.

Solution

y

x

y

x

10

Example Sketch the graph each of the following functions. State the domain and range of each function.

a) 32)( xxg b) 222)( xxh

Solution a) 3)2(32)( xxxg

xxfxf )(,3))2((

,2gD and ,3gR .

b) )2(22222)( xxxh

xxfxf 2)(;)2()2(

,2hD and ,2hR .

y

x

y

x

11

Example Sketch the graph each of the following functions by indicating the intercepts if any. State the domain and range of each function.

a) 148)( xxg b) 224)( xxh

Solution a) 1)2(21)2(4148)( xxxxg

xxfxf 2)(;1)2((

intercept 122)0( yg

intercept 4714801480)( xxxxxg

2,gD and ,1gR .

y

x

12

b) 2214224)( xxxh

22122

212 xx

xxfxf 2)(;221

21,hD and 2,gR .

intercept no has22)0( yRh

intercept 234242240)( xxxxxh

y

x

13

Example Sketch the graph each of the following functions by indicating the intercepts if any.Hence, state the domain and range of each function.

a) xxg 122)( b) 631)( xxh

Solution a) )2()1(2122)( xxxg

xxfxf 2)(;)2()1((

intercept 4)0( yg

intercept no has Impossible110)( xxxg

1,gD and 2,gR .

y

x

14

b) 1)2(31)2(3631)( xxxxh

xxxfxf 33)(;1))2(((

intercept no has61)0( yRh

intercept 371631630)( xxxxxh

2,gD and ,1gR .

y

x

.15

Modulus Functions (Absolute Value Function)

The basic modulus function is 0,)( axaxf and the graph is as shown below.

RD f and ,0fR .

Example Sketch the graphs of xaxf )( for 1a , 2a , 3a , 21a and

41a .

y

x

y

x

16

Example How to obtain the graph of the following functions using the graph of the basic function xxf 2)( .

State the domain and range of each function.

a) 212)( xxg b) 324)( xxh

Solution RD f and ,0fR .

a) 2))1((2)1(212)( xfxfxxg

Shift the graph to the left 1 unit and then upward 2 units. RDg and ,2fR .

b) 3223)2(2324)( xxxxh

)3()2()3(22 xfx

.

Shift the graph to the right 2 units and then downward 3 units. RDh and ,3hR .

17

Example Sketch the graph each of the following functions by indicating the axes intercepts if any. State the domain and range of each function.

a) 12)( xxg b) 322)( xxh

Solution a) 1)2(12)( xxxg

xxfxf )(;1))2((

intercept 1)0( yg

intercepts 1,30)( xxxg

RDg and 1,gR .

y

x

18

Solution b) 3223)2(2322)( xxxxh

xxfxf 2)(;32

intercept 7)0( yh

intercept no has3220)( ximpossiblexxh

RDh and 3,hR .

y

x

19

Rational funtions

The basic rational function is 0,)( axaxf and the graph is as shown below.

}0{\RD f and }0{\RRf .

The x-axis is the horizontal asymptote.

The y-axis is the vertical asymptote.

Example Sketch the graphs of xaxf )( for 1a , 2a and

21a .

y

x

y

x

20

Example How to obtain the graph of the following functions using the graph of the basic function

xxf 2)( .State the domain and range of each function.

a) 21

2)(

x

xg b) 243)(

xxxh

Note : For a rational function )(xf , )()( xfxf , we say reflect about one axis.

Solution }0{\RD f and }0{\RR f .

a) )2())1((2)1(21

2)(

xfxfx

xg

Shift the graph to the left 1 unit and then downward 2 units.

}1{\ RDg and }2{\ RRg .

b) 2

22

)2(32

2)2(3243)(

xxx

xx

xxxh

3)2(32

22

23

xfxx

.

Shift the graph to the right 2 units and then upward 3 units.

}2{\RDh and }3{\RRh .

21

Example Sketch the graph each of the following functions by indicating the axes intercepts if any. Also indicate the horizontal and vertical asymptotes. State the domain and range of each function.

a) 383)(

xxxg b)

42611)(

x

xxh

Solution a) 33

13

1)3(3383)(

xxx

xxxg

x

xfxf 1)(;3))3((

intercept 38)0( yg

intercept 380)( xxxg

}3{\ RDg and }3{\RRg .

Note : For a rational function )(xf , )()( xfxf , we say reflect about one axis.

y

x

22

Solution b) 3)2(2

142

1)42(342

611)(

xx

xx

xxh

x

xfxfx

21

21

)(;323)2(

intercept 4

11)0( yh

intercept 6

1106110)( xxxxh

}2{\RDh and }3{\ RRh .

y

x

23

Exponential Function

For exponential function xexf )( where

......718281828.2e , with the graph as shown below.

RD f and ,0fR .

The x-axis is the horizontal asymptote.

y

x

24

Example How to obtain the graph of the following functions using the graph of the basic function xexf )( .

State the domain and range of each function.

a) 2)( 1 xexg b) 3)( 2 xexh

Solution RDf and ,0

fR .

a) )2()1(2)1(2)( 1 xfxfexg x

Shift the graph to the right 1 unit and then downward 2 units. RDg and ,2gR

b) 3))2((3)2(3)( 2 xfxfexh x .

Shift the graph to the left 2 units and then upward 3 units. RDh and ,3hR .

25

Example Sketch the graph each of the following functions by indicating the axes intercepts if any. Also indicate the horizontal asymptote. State the domain and range of each function.

a) 23)( xexg b) 2)( 1 xexh

Solution a) xx exfxfexg )(;3))2((3)( 2

intercept 3)0( 2 yeg

intercept 23ln30)( 2 xxexg x

RDg and 3,gR

y

x

26

b) xx exfxfexh )(;)2()1((2)( 1

intercept 2)0( yeh

intercept no has possiblenot 20)( 1 xexh x

RDh and 2,gR

y

x

27

Exercise :Could you recognise the graphs of xexf 2)( and xexg 2)( below?

2ln2 2 xee xx

y

x

28

Logarithmic Function

The logaritmic function xxf ln)( with the graph as shown below.

,0fD and RR f .

The y-axis is the vertical asymptote.

y

x

29

Example How to obtain the graph of the following functions using the graph of the basic function xxf ln)( .

State the domain and range of each function.

a) 2)1ln()( xxg b) 3)2ln()( xxh

Solution ,0fD and RR f .

a) )2()1(2)1(2)1ln()( xfxfxxg

Shift the graph to the right 1 unit and then downward 2 units. ,1gD and RRg

b) 3))2((3)2(3)2ln()( xfxfxxh .

Shift the graph to the left 2 units and then upward 3 units. ),2( hD and RRh .

30

Example Sketch the graph each of the following functions by indicating the axes intercepts if any. Also indicate the vertical asymptote. State the domain and range of each function.

a) )2ln(3)( xxg b) 2)1ln()( xxh

Solution a) xxfxfxxg ln)(;3))2(()2ln(3)(

intercept 2ln3)0( yg

intercept 23)2ln(0)( 3 xexxxg

,2gD and RRg

y

x

31

b) xxfxfxxh ln)(;)2()1((2)1ln()( intercept 2)0( yh

intercept 12)1ln(0)( 2 xexxxh

RDh and 2,gR

y

x

32

Exercise Could you recognise the graphs of xxf 2ln)( and xxg ln2)( .

0,202

2ln22ln 2

xxx

xxxx

y

x

33