2.01 add math module 01 functions papers add maths/function...3 1.1 functions express the relation...
TRANSCRIPT
FUNCT
Organi
Jabatan Pelajaran
ADDITMATHEM
MOD
http://mathsmoz
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IONALATICS
IONS
zed by
Pulau Pinang 2006
ULE 1
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1
CHAPTER 1 : FUNCTIONS
Contents Page1.1 Concept map 2
1.2 Determine domain , codomain , object,image and range of relation
3
1.3 Classifying the types of relations 3
2.1 Recognize functions as a special relation.2.2 Expressing functions using function
notation.2.3 Determine domain , object , image and
range
4 - 5
3.0 Composite Functions 6 -9
4.0 SPM Questions 9 – 10
5.0 Assessment test 11 – 12
6.0 Answers 13 – 14
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CONCEPT MAP
Functions
FunctionNotation
CompositeFunctions
InverseFunctions
Relations
Typeof
relation
f: x y
Or………………
Objectimages
………….,,……………
……………
FUNCTIONS
One toone
Many toone
fg ( x ) = …………….
……….. Object
f ( x ) = y ………………
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1.1 Functions
Express the relation between the following pairs of sets in the form of arrow diagram,ordered pair and graph.
Arrow diagram Ordered pair Grapha ) Set A = Kelantan, Perak ,
Selangor
Set B = Shah Alam
, Kota Bharu ,Ipoh Relation: ‘ City of thestate in Malaysia ‘b )Set A= triangle,rectangle,
pentagon Set B = 3,4,5 Relation : ‘ Number ofSides’
1.2 Determine domain , codomain , object, image and range of relation.
List down the domain , codomain , objects , images and the range of the following relation.
Domain = ……………………………………… Codomain = ……………………………………… Object =……………………Image =……………………Range =…………………...
3
2
-2
5
4
3
9
Set P Set QDiagram 1
-3 1
1
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1.3 Classifying the types of relations
State the type of the following relations
a)
…………………………………
c)
…………………………………
2.0 Functions2.1 Recognize functions as a spe2.2 Expressing functions using f2.3 Determine domain , object , i
2.1 Identify each of the following
a)
……………………………
b)
…
x
2
4
6
AB
p
q
r
1
2
3
3
24
9-3
x X2
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…………..
b )
………………………………………….
…………..
d)
……………………………………………
cial relation.unction notation.mage and range
relations is a function or not.
4
16
2
x
36
4
9
Prime
-3
x
3
24
-29
-3
Even
xX
Type of number
c)A B
p
q
r
a
b
c
A B
a
b
c
p
q
r
……
ma
4……………
thsmozac.b
……… ………
d
logspot.com
……………
………2.2 Express each of the following functions using function notation.
a)
Functf : xor f ( x
b ) c)
2.3 a)
( i ) f
f
………
b ) Fin
i )f : xobject
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ion notation……………) = …………
Find the imag
: x 2x + 9
(5 ) =
……………
d the object f
2x – 3 , fiwhen the im
A
2
3
4
B
4
6
A B
2
3
4
9
A B
1
2
5
7
..…
Functig : xor g (
e for each of the f
……
( ii ) f
f
………
or each of the fol
nd theage is 5. ii )f : x
object
8
http://ma
5on notation……………
x ) = ………
ollowing fun
: x 2
35 x
(-3 ) =
……………
lowing functi
3
82 x,
when the im
4
thsmozac.b
…
..…Functih : xor h (
ctions.
……
iii) h :
fin
………
ons.
find the
age is 3.
i )f : x
object
16
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on notation……………
x ) = …………
x5
x+6
d h ( -2 )
……………
x
6– 7 , fi
when the im
3
n
a
..…
……..
d the
ge is -5.
9
6
c ) Find the value of x for each of the following function.
i ) f ( x ) = 2x + 7for which f ( x ) = 3
ii ) g ( x ) =12
8
xfor which g (x) = 4
iii) h(x) =2
3x
for which h ( x ) = x
3.0 Composite Functions
3.1 ( a ) Find the value for each of the following composite functions
i ) f( x ) = x + 2 andg ( x) = 5x + 3find fg ( 2 ) =
ii ) g ( x ) = 2 +5x
find g2(4)
iii) f(x) = 3x+2
1and g ( x )
=1
1
x.Find fg ( 1 )
acb
fg
f: a bg: b c
gf : a c
f g
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( b ) Find the following composite function.
( i ) f( x ) = 2x + 3g( x ) = 1 – x
fg ( x ) =
( ii ) f( x ) = 2x + 3g( x ) = 2 + 5 x2
gf ( x ) =
( iii ) f ( x ) = 1 -2
x
g ( x ) =x
4
fg ( x ) =
( c) Find the value of x for each of the following composite function.
i) f ( x ) =x
3
g ( x ) = 2x + 1
fg ( x ) = 5
ii )f( x ) = 2x + 4
g ( x ) = x -2
fg ( x ) = 2
iii) f ( x ) = 1 -2
x,
g (x ) =x
4
fg ( x ) = -1
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( d ) Solve
i )Given the functionf: x 4x + k ,g : x x – 2fg : x mx + 8
Find the value of k and m
ii )Given the functionf: x 9 – 2x ,g : x ax + b andfg: x 1 – 6x
Find the value of a and b.
iii)Given the functionf: x 2x – 1 ,g : x 4x andgf : x ax + b
Find the value of a and b
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4.0 SPM QUESTIONS
SPM 2004 ( paper 1, question no 1 )
1. Diagram 1 shows the relation between set P and set Q
State( a ) the range of the relation,( b ) the type of relation. [2 marks]
Answer:
d
e
f
x
y
z
w
Set P Set Q
Diagram 1
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SPM 2004 ( paper 1, question no 3)
1. Given the function h ( x ) =x
6, x 0 and the composite function hg ( x ) = 3x , find
( a ) g ( x )( b ) the value of x when gh ( x ) = 5. [ 4 marks]
Answer:
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SPM 2005 Question
1. In Diagram 1 , the function h maps x to y and the function g maps y to z.
Determine( a ) gh ( 2 )
Answer:
y zx
2
5
8
h g
Diagram 1
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5.0 Assessment( 30 minutes)
1)
The diagram above shows the relation between set A and set B. Statea) the type of relationb) the range of relation
Answer:
2) Given that f : x2x + 7 find the object when image is 3.Answer:
3) Given that f ( x ) = 10 – kx and f ( 2 ) = 4 ( k constant ) . Find the value of k.Answer:
a
b
c
q
r
s
P
ABDiagram 1
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4) Given that f ( x ) = 4x -1 and g ( x ) = 2x + 3 .Findi ) fg ( x )ii ) fg ( - 2 )
Answer:
5 ) Given the function f : x px + 2 and g : x qx + 3 . If the composite function fg issuch that fg ( x ) = 8x + 8 , find the values of p and q.Answer:
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6.0 ANSWERS
1.1
Arrow diagram Ordered pair Graph
(kel, kb), (Sel,
S.Alam), (Perak,Ipoh)
(Tri, 3), (rec,
4), ( Pen,5)
1.2 Domain = -3,-2,1,2,3 , codomain = 1,3,4,5,9 ,
Image = 1,3,4,5,9 , Range = 1,4,9 1.3 ( a ) one to one ( b ) One to many( c ) many to one( d ) many
2.1 ( a ) function ( b ) Not function ( c ) function
2.2 (a) 2x (b) x2 (c) 2x + 3
2.3 a) ( I ) 19 (ii) -6 (iii ) 3 b) (i) 2 (ii)2
1(iii) 3 c
3.1 a) ( i) 15 (ii ) 102 ( iii) 2
b) (i) 5 – 2x (ii) 20x2+60x+47 (iii) 1-x
2
c) (i) -5
1(ii) 1 (iii) 1
d) (i) k=16,m=4 (ii) a=3,b=4 (iii) a=8,b= - 4
4.0 SPM QUESTIONSSPM 2004( P1,Q1) a) range = x,y b )many to one
SPM 2004( P1,Q3) (a) g(x) =3
2, x 0 ( b) x = 15
SPM 2005( P1,Q1) 8
Tri
rec
Pen
4
5
3
AB
Kel
Sel
PerIpoh
SA
KB
AB
KB
SA
Ipoh
3
3
4
5
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Object =1,2,
to many
) (i) -2 (ii)
Kel Sel Per
recPen
3,-2,-3
2
1(iii) -3
Tri
15
5.0 Assessment( 30 minutes)
1) ( a)many to many ( b ) range = p,q,r
2) -2
3) k = 3
4) (i) 8x+11 (ii) -5
5) p =2
1, q = 16
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FUNCT
ADDITMATHEM
MOD
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IONALATICS
6
IONS
ULE 2
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CHAPTER 1 : FUNCTIONS
Contents Page
1.0 Inverse Functions ( concept map ) 2 – 4
2.0 Absolute Function 4 - 6
3.0 SPM Questions 7 – 8
4.0 Assessment ( 30 minutes ) 9 – 10
5.0 Answers 11 – 13
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CHAPTER 1 : FUNCTIONS
1.0 Inverse Functions
1.1 Determine the object by inverse mapping1.2 Determine the inverse functions
a ) Find the inverse function of each of the following functions.
i ) f ( x ) = x + 3 ii) f ( x ) =5
xiii ) f ( x ) =
2
13 x
Functions
FunctionNotation
CompositeFunctions
InverseFunctions
Relations
Typeof
relation
f: x y
Orf(x) = y
f ( x ) = y f-1( y ) = x
ObjectimagesDomain
Codomain ,Range
FUNCTIONS
Arrow diagram
Ordered pairs
graph
One toone
One tomany
Many toone
Manyto many
fg ( x ) = f [ g(x) ]
image Object
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iv ) f ( x ) = 7 – 5x v ) f ( x ) =4
3 xvi ) f( x ) =
3
45 x
b ) Find the inverse function of each of the following functions in terms of p and q
i ) f ( x ) = px - q ii ) f ( x ) =q
px iii) f ( x ) = px +
q
1
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c ) Given the function f : x1
2
x
x, x 1 and g(x) =2x -6 , find f-1 g .
Answer:
d ) Inverse function f is define by f-1 : x 12
5
x
x, x
2
1. Find f ( 2 )
Answer:
2.0 Absolute Function
1. Sketch the graph of each of the following functions
a ) f ( x ) = x b ) f ( x ) = x + 1
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c) f (x ) = | x | d ) f ( x ) = | x + 1 |
e ) f (x) = |x| + 1 f ) f ( x ) = | x | - 1
2.Sketch the graph of each the following functions and state the corresponding range.
a) f : x 2x – 3 for 0 x 4
Range :……………………………….
b) f : x |2x – 3| for 0 x 4
Range : …………………………………………
f(x) f(x)
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c) f : x | 5 – 2x |for -1 x 4
Range :……………………………….
d) f : x |9 – 2x| for 0 x 6
Range :……………………………….
e) f : x |2x| – 1 for -1 x 3
Range :……………………………….
f ) f : x | 3x | for -2 x 2
Range :……………………………….
f(x)
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3.0 SPM QUESTIONS
SPM 2004 Question.
1. Given the functions h : x 4x + m and h-1 : x 2kx +8
5, where m and k are
constants, find the value of m and of k.[ 3 marks]
Answer:
SPM 2005 ( Paper 1, Question 1 )
2. In Diagram 1 , the function h maps x to y and the function g maps y to z.
Determine( a ) h-1 ( 5 )
Answer:
y zx
2
5
8
h g
Diagram 1
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SPM 2005 ( Paper 1, Question 2 )
1.The function w is defined as w ( x ) =x2
5, x 2.
( a ) w-1 ( x ),( b ) w-1 ( 4 ). [ 3 marks]
Answer:
SPM 2005 ( Paper 1, Question 3 )1.The following information refers to the functions h and g.
Find gh-1 ( x ). [ 3 marks]
h: x 2x – 3
g: x 4x - 1
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4.0 Assessment ( 30 minutes )
1.A function f is defined by f: x 6 -2
1x
Finda ) f ( x ) b ) f-1 ( 5 )
Answer:
2. Inverse function f is defined by f-1 : x 3
45 x
find f ( 2 ).
Answer:
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3.Given the function f : x 2x - m and inverse function f-1 : x nx +3
7
Find the value of m and n.
Answer:
4. Sketch the graph of the function f ( x ) = |2x – 5 | for 0x6. Hence , state thecorresponding range.
Answer:
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5.0 ANSWERS1.1
( a )( i) x- 3 ( ii) 5x (iii)3
12 x(iv)
5
7 x(v) 3 – 4x (vi)
4
35 x
( b )p
qx (ii ) xq + p (iii)
p
x-
pq
1
( c)x
x
27
42
, x
2
7
( d ) -1
Absolute function
2.
.8)(3 xf 0 f(x) 5
f(x)
x
b )
_ 5
|1
|2
f(x)
x
a )
_ 8
|1
|2
_-3
f(x)
x
f )
_ -1
|-1
|1
f(x)
x
e )
_ 1
|-1
f(x)
x
d )
_ 1
|-1
f(x)
x
c )
f(x)
x
b )
_ 1
|-1
f(x)
x
a )
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0 f(x) 7 0 f(x) 9
-1 f(x) 5 0 f(x) 6
SPM 2004 ( P1,Q2)
K=8
1, m=-
2
5
SPM 2005 ( P1,Q2)(a) 2SPM 2005 ( P1,Q2)
( a )x
x 52 , x 0 ( b )
4
3
SPM 2005 ( P1,Q3)2x+ 5Assessment ( 30 minutes )1)( a) 12 – 2x ( b ) 8
2) -4
1
3) n =2
1, m=7
4)
f(x)
x
f )_ 6
|2
|-2
f(x)
x
d )
_ 9
|4
|5
f(x)
x
e )_ 5
|1
|-1 _ -1
_ 1
|3
f(x)
x
c )_ 7
|2
|3
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The corresponding range of f(x
f(x)a )
_ 5
|2
|3
_ 7
|6
http:
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) = 0 f(x) 7
x
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