Download exercise to practices and more Exercises Circuit Theory in PDF only on Docsity!
1 FUNCTIONS
1.0 RELATIONS
Notes :
(i) Four types of relations : one-to-one, many-to-one, one-to-many and many-to-many. (ii) Three ways to represent relations : arrowed diagram, set of ordered pairs and graph. (iii) The arrow diagram representation is of the form below :
In the above relation, b is the imej of a ; a is the object of b.
(iv) Set of ordered pairs of a relation takes the form of { (a , b), (c , d), ….. }
EXAMPLE QUESTION
A relation from P into Q is defined by the set of
ordered pairs
{ (1, 4), (1, 6), (2, 6), ( 3 , 7) }.
State
(a) the image of 1, Ans: 4 and 6 (b) the object of 4, Ans: 1 (c) the domain, Ans: { 1, 2, 3 } (d) the codomain, Ans: {2,4,6,7,10} (e) the range, Ans: { 4, 6, 7 } (f) the type of relation Ans: many-to-many
A relation from P into Q is defined by the set of ordered pairs { (2, 2), (4, 2 ), (6, 7) }.
State (a) the image of 4, Ans: (b) the objects of 2 , Ans: (c) the domain, Ans: (d) the codomain, Ans: (e) the range, Ans: (f) the type of relation Ans:
a is the object
b is the image
P Q
a (^) b
P :Domain
Q : Codomain
A B
The relation from A into B above can also be written as set of ordered pairs {(1, 4), (3, 6) }
4 is the image of 1 ; The Object of 6 is 3.
_
_
_
_
A : Domain (^) B : Codomain
Set { 4, 6 } is the RANGE for the mapping from A to B
P = { 1, 2, 3} Q = {2, 4, 6, 7, 10}
P = { 1, 2, 3} Q = {2, 4, 6, 7, 10}
P = { 2, 4, 6} Q = {2, 3, 6, 7, 10}
1.1 FUNCTION NOTATIONS
Notes:
(i) A function is usually represented using letters in its lower case : f , g , h …..
(ii) Given the function (^) f : x 2 x 1 , we usually write it in the form f( x ) = 2 x + 1 before answering
any question.
1.2 FINDING THE VALUE OF FUNCTIONS [Mastery Exercise]
f ( a ) represent (a) the value of f (x) when x = a. (b) the image of f when the object is a
EXAMPLE QUESTION
- Given the function f : x 2 x 1 ,
find (i) f (3) (ii) f (-4)
Answer : f(x) = 2 x + 1
(i) f ( 3 ) = 2 ( 3 ) + 1
= 6 + 1
(ii) f ( - 2 ) = 2 (- 2 ) + 1
= - 4 + 1
- Given the function f : x 2 x 3 ,
find (i) f (2) (ii) f (-1)
Answer : f(x) = 2 x + 3
(i) f( 3 ) = 2( ) + 3
=
(ii) f( - 1 ) = 2 ( ) + 3
=
- Given the function : 3 2 g x x , find
(i) g(0) (ii) g(4)
Answer : g (x) = x
2
(i) g ( 0 ) = 0
2
= 0 - 3
(ii) g ( 4 ) = 4
2
= 16 – 3
- Given the function : 5 2 g x x , find
(i) g(0) (ii) g(2)
Answer : g (x) = x
2
(i) g ( 0 ) =
=
(ii) g ( 2 ) =
=
- Given the function 3 4
x
h x , 3
x ,
find h (2).
Jawapan : 3 4
x
h x
h ( 2 ) = 3 ( 2 ) 4
- Given the function 2 3
x
h x , 2
x ,
find h (3).
Jawapan: h ( x )
h ( 3 ) =
3
1.3 PROBLEM OF FUNCTIONS INVOLVING SIMPLE EQUATIONS. [1]
Note : Questions on functions usually involve solving linear / quadratic equations.
EXAMPLE QUESTION
- Given that f : x 2 x 1 , find the value of x
if f(x) = 5.
Answer : f (x) = 2 x – 1
2 x – 1 = 5
2 x = 6
x = 3
- Given that f : x 4 x 3 , find the value of x if
f ( x ) = 17.
Answer : f (x) = 4 x – 3
4 x – 3 = 17
4 x =
x = 5
- Given that f : x 5 x 3 , find the value of x
if f ( x ) = -7.
Answer : f (x) = 5 x + 3
5 x + 3 = -
5 x = - 10
x = - 2
- Given that f : x 3 x 7 , find the value of x if
f ( x ) = 4.
Answer :
x = - 1
- Given that g : x x 2 , find the value of y
if g(y) = 2y - 3
Answer : g(x) = x + 2
g(y) = y + 2
y + 2 = 2 y – 3
3 + 2 = 2 y - y
y = 5
- Given that g : x 3 x 2 , find the value of y if
g(y) = 2y + 4.
Answer :
y = 6
- Given that f : x 7 x 3 , g : x 4 x 15 ,
find the value of p if f(p) = g(p).
Answer : f(x) = 7x – 3, g(x) = 4x + 15
f(p) = 7p – 3, g(p) = 4p + 15
f(p) = g(p)
7p – 3 = 4p + 15
3p = 18
p = 6
- Given that f : x 2 x 10 , g : x 4 x 2 ,
find the value of x if f(x) = g(x).
x = - 6
PROBLEMS OF FUNCTIONS INVOLVING SIMPLE EQUATIONS. [2]
EXAMPLE QUESTION
- Given that f : x 2 x 10 , g : x x 7 ,
find the value of x if f(x) = 2 g(x)
A: f(x) = 2x - 10 , g(x) = x – 7
2x – 10 = - 2 (x – 7)
2x - 10 = - 2x + 14
4x = 24
x = 6
- Given f : x 3 x 9 , g : x x 7 , find the
value of x if f(x) = 3 g(x).
x = 5
- Given that f : x 3 x 4 , find the values
of x if f (x) = x
2 .
A: f(x) = 3x + 4
x 2 = 3x + 4
x
2
(x + 1) (x – 4) = 0
x = -1 or x = 4
- Given that f : x 2 x 8 , find the values of x if
f (x) = x
2 .
x = 4 , - 2
- Given that : 3 6
2 g x x , find the values of
y if g (y) = 6.
A: g(x) = 3x 2
g(y) = 3y
2
3y
2
3y 2
(÷3) : y 2
(y + 2) (y – 2) = 0
y = -2 or y = 2
- Given that : 2 5
2 g x x , find the values of y if
g (y) = 45.
x = - 5 , 5
- Given that : 3 6 2 g x x , find the values of
p if g (p) = 7p.
A: g(x) = 3x 2
g(p) = 3p 2
3p
2
3p
2
(3p+2) (p – 3) = 0
p = 3
atau p = 3.
- Given that : 2 3 2 g x x , find the values of p if
g( p ) = - 5p.
p = 1/2 , - 3
1.4 FINDING THE VALUE OF COMPOSITE FUNCTIONS [Mastery Exercises]
Notes : To find the value of fg(a) given the functions f and g, we can first find fg(x) and then
substitute x = a. We can also follow the EXAMPLES below.
EXAMPLE 1 :
Given that f : x 3 x 4 and g : x 2 x ,
find fg(3).
Answer : f(x) = 3x - 4 , g(x) = 2x
g(3) = 2(3)
fg(3) = f [ g(3) ]
= f ( 6)
= 3 (6) - 4
EXAMPLE 2 :
Given that f : x 3 2 x and 2 g : x x , find
gf(4).
Answer : f(x) = 3 – 2x , g(x) = x 2 .
f(4) = 3 – 2(4)
gf(4) = g (-5)
2
EXERCISES
- Given that f : x 2 x 1 and g : x 3 x ,
find f g(1).
- Given that f : x 2 x 9 and g : x 1 3 x ,
find gf(3).
- Given the functions f : x x 3 and
g : x 4 x 1 , find
(a) f g(1) (b) gf(1)
- Given the functions f : x 3 x 7 and
g : x 4 2 x , find
(a) f g(0) (b) gf(0)
FINDING THE VALUE OF COMPOSITE FUNCTIONS [Reinforcement Exercises]
- Given that f : x 2 x 3 and g : x 4 x ,
find fg(2).
- Given that f : x 2 x 5 and g : x 5 x , find
gf(7).
- Given that f : x x 4 and g : x 2 x 1 ,
find
(a) fg(1) (b) gf(1)
- Given that f : x 3 x 4 and g : x 5 2 x ,
find
(a) fg(0) (b) gf(0)
5 Given the functions : 3 1 2 f x x and
g : x 2 x , find
(a) fg(-1) (b) gf(-1)
6 Given that f : x 3 x and 2 g : x 2 x , find
(a) fg(-2) (b) gf(-2)
- Given the functions f : x x 2 and
2 g : x 2 3 x x , find
(a) fg(1) (b) gf(1)
- Given the functions f : x 2 x and
2 g : x 1 4 x 3 x , find
(a) fg(-1) (b) gf(-1)
1.6 BASIC EXERCISES BEFORE FINDING THE COMPOSITE FUNCTIONS [ 1 ]
EXAMPLE 1 :
Given f : x 3 x 1 ,
f(x) = 3x + 1
thus (a) f(2) = 3(2) + 1
(b) f(a) = 3 a + 1
(c) f(p) = 3 p + 1
(d) f(2k) = 3 (2k) + 1 = 6k + 1
(e) f(2x) = 3 (2x) + 1 = 6x + 1
(f) f(x 2 ) = 3 x 2
EXAMPLE 2 :
Given g : x 5 4 x ,.
g(x) = 5 – 4x
thus (a) g(2) = 5 – 4(2) = 5 – 8 = -
(b) g(a) = 5 – 4a
(c) g(p) = 5 – 4p
(d) g(3k) = 5 – 4(3k) = 5 – 12 k
(e) g(x
2 ) = 5 – 4x
2
(f) g (3+2x) = 5 – 4 (3+2x)
= 5 – 12 – 8x
= - 7 – 8x
EXERCISES
- Given f : x 2 x 3 ,
f(x) = 2x + 3
thus (a) f( 2 ) = 2(2) + 3 =
(b) f(a) =
(c) f(p) =
(d) f(2k) =
(e) f(x
2 ) =
(f) f(-x
2 ) =
(g) f( - 3x) =
2 Given g : x 2 4 x.
g(x) =
thus (a) g (2) =
(b) g(a) =
(c) g(s) =
(d) g(3x) =
(e) g(x
2 ) =
(f) g (3+2x) =
(g) g(2 – 4x) =
BASIC EXERCISES BEFORE FINDING THE COMPOSITE FUNCTIONS [ 2 ]
- Given f : x 4 2 x ,
f(x) = 4 – 2x
thus (a) f( 3 ) = 4 – 2 ( 3 ) =
(b) f(-x) =
(c) f(2+x) =
(d) f(3 - x) =
(e) f(x 2 ) =
(f) f(-x
2 +2) =
2 Given g : x 2 x 1.
g(x) =
thus (a) g(-1) =
(b) g(2x) =
(c) g(x-2) =
(d) g(-3x) =
(e) g(x 2 ) =
(f) g (1-2x) =
- Given f : x 1 x ,
f(x) =
thus (a) f(10) =
(b) f(3x) =
(c) f(2 - x) =
(d) f(4+x) =
(e) f(2x- 3) =
(f) f(x
2 ) =
4 Given g : x 2 ( x 2 ),.
g(x) =
thus (a) g(-1) =
(b) g(2x) =
(c) g(x-1) =
(d) g(-x) =
(e) g(x 2 ) =
(f) g (1+2x) =
REINFORCEMENT EXERCISES FOR FINDING COMPOSITE FUNCTIONS
- Given the functions f : x 3 x 2 and
g : x 2 2 x , find
(a) fg(x) (b) gf(x)
- Given that f : x 3 x and 2 g : x 2 x , find
the composite functions
(a) fg (b) gf
- Given the functions f : x x 4 and
g : x 1 3 x , find
(a) fg(x) (b) gf(x)
- Given that f : x 3 x and g : x 4 x 3 ,
find the composite functions
(a) f g (b) gf
- Given the functions f : x x 2 and
g : x 5 2 x , find
(a) fg(x) (b) gf(x)
- Given that f : x 2 5 x and 2 g : x 1 x ,
find the composite function gubahan
(a) fg (b) gf
REINFORCEMENT EXERCISES FOR FINDING COMPOSITE FUNCTIONS [ f 2 and g 2 ]
- Given that f : x 3 x 2 , find f
2 ( x ).
f(x) = 3x + 2
f 2 (x) = f [f(x)]
= f (3x + 2) or 3f(x) + 2
= 3(3x+2) + 2
=
- Given that f : x 3 x , find the function f
2 .
- Given that g : x 1 3 x , find gg ( x ) 4. Given that g : x 4 x 3 , find the function
g 2 .
- Given that f : x x 2 , find ff ( x ). 6. Given that f : x 2 5 x , find the function f 2 .
- Given that f : x 3 4 x , find f
2 ( x ). 8. Given that f : x 5 x , find the function f
2 .
- Given that g x x 2
: , find g 2 ( x ). 10. Given that 2
x
h x , find the function f 2 .
DRILLING PRACTICES ON CHANGING SUBJECT OF A FORMULA [ 2 ]
- If y = 4
2 x 5 , then x =
2 x 5 = y
2x – 5 = 4y
2x =
x =
- If y = 5
3 x 2 , then x =
- If y = 5 + 6x , then x =
- If y = 4 – 2
x , then x =
- If y = 6 + 2
x , then x =
- If y = 6x – 9 , then x =
- If y = 3 2
x
, then x = 8. If y = 4 x
, then x =
- If y = 4 2
x
x , then x = 10. If y = x
x
3
, then x =
DETERMINING THE INVERSE FUNCTION FROM A FUNCTION OF THE TYPE
ONE-TO-ONE CORRESPONDENCE
EXAMPLE 1 :
Given that f(x) = 4x – 6 , find f
EXAMPLE 2 : Given that f : x x
x
, x 2,
find f
Answer :
Given f(x) = 4x – 6
so f (y) = 4y – 6
then f
f
- 1 ( x ) = y when x = 4y – 6
x + 6 = 4y
y = 4
x 6
f
x 6
Given f( x ) = x
x
so f( y ) = y
y
2
2 1
then f
y
2
2 1 ) = y
f
y
2
2 1
x (2 - y ) = 2 y + 1
2 x – xy = 2 y + 1
2 x – 1 = 2 y + xy
2 x – 1 = y (2 + x )
y = x
x
2
2 1
f
x
x , x -2.
EXERCISES
- Given that f : x 4 + 8x , find f
- 1 . 2. Given that g : x 10 – 2x , find g - 1 .
- Given that f : x 4 – 3x , find f
- 1 . 4. Given that g : x 15 + 6x , find g - 1 .
FINDING THE VALUE OF f
- 1
(a) WHEN GIVEN f(x)
Example : Given that f : x 2x + 1, find the value of f
[ Finding f
METHOD 2
[ Without finding f
f(x) = 2x + 1
f(y) = 2y + 1
f
f
- 1 (x) = y when x = 2y + 1
x – 1 = 2y
y = 2
x 1
f
x 1
f
f(x) = 2x + 1
Let f
f(k) = 7
2k + 1 = 7
2k = 6
k = 3
f
( If you are not asked to find f
- 1 ( x ) but only want to have the value of f - 1 ( a ), then use Method 2 )
EXERCISES
- Given that f : x x + 2, find the value of
f
- 1 (3). 2. Given that g : x 7 – x , find the value of
g
- Given that f : x 7 – 3x , find the value of
f
- 1 (-5). 4. Given that g : x 5 + 2x , find the value of
g
- Given that f : x 3x - 2, find the value of
f
- 1 (10). 6. Given that g : x 7 – x , find the value of
g
- Given that f : x 4 + 6x , find the value of
f
- 1 (-2). 8. Given that g : x 2
g
- Given that f : x 3
x
, find the value of
f
- 1 ( - 1 ). 10. Diberi g : x 2 x
, cari nilai g
Finding the Other Related Component Function when given a Composite Function
and One of its Component Function
TYPE 1 ( Easier Type )
Given the functions f and fg, find the function g.
OR
Given the functions g and gf, find the function f.
TYPE 2 ( More Challenging Type )
Given the functions f and gf , find the function g.
OR
Given the functions g and fg , find the function f.
EXAMPLE :
- Given the functions f : x 2 x 3 and
fg : x 6 x 1 , find the function g.
Answer : f(x) = 2x + 3
fg(x) = 6x – 1
Find g(x) from fg(x) = 6x – 1
f [ g(x) ] = 6x – 1
2 g(x) + 3 = 6x – 1
2 g(x) = 6x – 4
g(x) = 3x - 2
- Given the functions f : x 2 x 5 and
gf : x 10 x 25 , find the function g.
Answer : f(x) = 2x – 5
gf(x) = 10x – 25
Find g(x) from gf(x) = 10x – 25
g [ f(x) ] = 10x – 25
g ( 2x – 5) = 10x – 25
g ( 2y – 5) = 10y – 25
g(x) = 10y – 25 when x = 2y – 5
x + 5 = 2y
y = 2
x 5
So : g(x) = 10 ( 2
x 5 ) – 25
= 5x + 25 – 25
g(x) = 5x
EXERCISES
- Given the functions f : x 2 x 2 and
fg : x 4 6 x , find the function g.
Answer :
- Given the functions f : x 2 x 2 and
g f : x 5 6 x , find the function g.
Answer :
