Fjax0112 1314 nota2
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Transcript of Fjax0112 1314 nota2
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