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FUNCTION

A function is a set of ordered pairs which relates two sets such that to each element of one set there is only one element of the second set
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION



EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION

Example
Represent the following set of ordered pairs in a pictorial diagram.
(1,4), (2,3), (-1,2), (-2,-3).

Solution
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
A function whose graph is such that any line drawn parallel to the x – axis at any point cuts it at only one point is one-to-one function.

Examples.
1. Write the expression of a function ‘ double plus one’
Solution
f:x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION 2x +1
b)
2.Given the function G(x) = 4x-1. Find the value of G(-2).

Solution
G(x) = 4x -1
G(-2) = 4(-2) -1
G(-2) = -8 -1
G (-2) = -9
Exercise 2.1
1. Write each of the following function in the form f: x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION f(x) use any functions symbol to represent the functions
(a)Divide by 5 and add 2.
(b)Subtract 7 and square
(b)Cube and then double

Solution
(a) F:x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION f(x)
F:x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION + 2
(b) F: x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION (x-7)2
(c) F:x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION x3+2x

2. Find the value of the function for each given value of x.
(a) f(x)= 2x+3; when
(i) x=1
(ii) x= -2
(iii) x =a

Solution
(a) (i) when x=1

f(x)= 2x+3
f(1)= 2(1)+3
f(1)= 2+3
f(1)= 5.

(ii)when x= -2.

f(x) = 2(-2) +3
f(-2) = -4+3
f(-2) = -1

(iii) when x=a
f(x) = 2(a) +3
f(a) =2a+3
f(a) = 2a+3

(b) C(x) = x3 when
(i) x=1
(ii) x = -1
(iii)x= 0
(iv)x=b

Solution
(i) C(x) =x3
C (1) =13
C (1) =1
C (1) =1

(ii)C(x) =x3
C (-1) = (-1)3
C (-1) = -1


(iii)C(x) =x3
C (0) =03
C (0) =0

(iv)C(x) =x3
C(b) = b3

(c) K(x) = 3-x; when
(i) x= -1
(ii)x=7
Solution
(i) K(x) = 3-x
K (-1) = 3- (-1)
K (-1) = 4

(ii) k(x)=3 – x
k(7) = 3-7
k(7) = -4
DOMAIN AND RANGE OF FUNCTIONS
Example 1.
1. Find the domain and range of f(x) =2x+1
let f(x) = y
y=2x+1
Solution
Domain = {x: xEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTIONR}
Range – make x the subject
y=2x+1
y-1=2x
(y-1)/2 =x
x=(y-1)/2
Range={y: yEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTIONR}
b Example 2
If y =4x+7 and its domain ={x : -10 < x< 10} find the range.
Solution
Domain = {x: -10 < x< 10}
Range ={y : -33 < y < 47}
y = 4x + 7
Table of values
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
c Example 3.
Y= √x and domain is -5 < x < 5, Find its range.

Solution
Domain = {x: -5 < x < 5}
Range =y

Table of value of x = √(y)
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
(y)2= (√x)2
y2= x
x= y2
√5= √y2
y = √5
Range = {y: o < y < √5}
d Example 4.
Given F(x) =EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION find the domain and range.
Solution
Let f(x)=y
Domain of y= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
For real values of y: 1-x2 > 0
-x2 > 0-1
-x2 > -1
x2 < 1
√x2< √1

X <√1
X <1
Domain {x : X <1 }
let F(x) = y
y=EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION

To get the range, make x the subject
y2 = (EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION)2
y2= 1-x2
X2=1-y2
Therefore EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
X=EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
For real value of x:
1- y2 > 0
y2 < 1
y < √1
y < 1
...Range ={y : y <1}
LINEAR FUNCTIONS
Is the function with form f(x) = mx + c.

Where:
f(x)= y
m and c are real numbers
m is called gradient[ slope].
c is called y – intercept.
Example
1. Find the linear function f(x) given the slope of -2 and f( -1)=3
Solution

Given:
m(slope)=-2
x=-1
f(-1)=3
from;
f(x) =mx +c
f(-1) =( -2x-1)+c
3 =2+c
3-2 =c
c=1
f(x) = -2x+1

Example 2.
Find the linear function f(x) when m=3 and it passes through the points (2, 1)
Solution
f(x) =mx +c
f(2) = 3(2)+ c
1= 6+c
1-6 =c
-5 = c
c= -5
f(x) = 3x + -5
f(x) =3x-5

Example 3
Find the linear function f (x) which passes through the points (-1, 1) and (0, 2 )
Solution
Slope = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
=1
m = 1
f(x) = mx +c
f(-1) = 1x(-1) +c
1 =-1+c
c=2
f(x) = 1x +2
f(x) = x +2.

4.Draw the graph of h(x) = 3x-4
Table of values of function
X
-1
0
1
h(x)
-7
-4
-1
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION

Exercise
In problems 1 to 3 find the equation of a linear function f(x) which satisfies the given properties. In each case, m dissolves the gradient.
1). m =-3, f(1) = 3
2). m=2, f(0) =5
3). f(1) =2, f(-1) =3

4. Givenm= -4 , f(3) = -4 Find f(x)

In the problem 5 to 9 draw the graphs of each of the given functions without using the table of values
5) f(x)= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION+EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION

6) f(x) =4


Solution
1. f(x) = mx + c

f(x) = -3(1)+c
f(x) =-3 +c
3= -3 +c
3+3 =c
6=c
C=6
f(x) = -3x +6

2. f(x) =mx +c
f(0) = (2 x 0)+c
5= 0+c
c=5
f(x) =2x+5
3)
3.f(1) =2, f(-1) =3 Alternatively
m = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION f(-1) = m(-1) + c
M= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION 3= -m + c…………………..(i)
f(1)=2 f(1) = m(1) + c


f(x)= mx +c 2= m + c(ii)
f(1)= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTIONx 1+c Solve (i) and (ii) Simultaneously
f(1) = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION+c -m + c = 3
2=EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION +c + m + c = 2
2+EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION= c C=5/2
C=2EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION put c in (i)
f(x) =EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION+ 2EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION 3 = -m + 5/2
m = 5/2 – 3
m= -1/2
...f(x) =EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION+ 2EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION

4.
f(x)= mx + c
f(x) = -4(3)+c
-4 =-12+c
-4+12= c
8=c
C=8
f(x) = -4x+8

5. f(x)= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION+EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
y- intercept, x=0
f(0)= 2/5[0] +1/5
f(0)= 0+1/5
y=1/5
{0,0.2}
x- intercept, y=0
0=2/5[x] +1/5
x intercept = – ½ ( -1/2, 0)
y intercept = 1/5 ( 0,0.2)
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
6. f(x) =4

From f(x) = y
y = 4
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
QUADRATIC FUNCTIONS
A quadratic function is any function of the form;
f(x)=ax2+bx+c
Where a ≠ 0
a ,b and c are real numbers.
When a = 1, b=0, and c= 0
X
-3
-2
-1

0
1
2
3
f(x)
9
4
1
0
1
4
9
The shape of the graph of f(x)=ax2+bx+c is a parabola
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
•The line that divides the curve into two equal parts is called a line of symmetry [axis of symmetry]
•Point (0,0) in f(x)=x2 called the turning point (vertex).
If “a” is positive the turning point is called minimum point (least value).
If “a” is negative the turning point is called maximum point.
PROPERTIES OF QUADRATIC FUNCTIONS
Think of f(x)= ax2+bx+c
y=a(x2+bx/a)+c
y= a( x2+bx/a+b2/4a2)+ c-b2/4a2
=a(x + EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION) 2 +EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
x> 0 then
a(x+EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION)2 > 0
y = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
This is when x = –EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
The turning point of the quadratic function is EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
Example
Find the minimum or maximum point and line of symmetry f(x) = x2 – 2x-3. Draw the graph of f(x)
Solution:
Turning point = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
= ( – EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION)

Maximum point = (1, -4)

Line of symmetry is x=1
x
-5
-2
-1
0
1
2

3
4
f(x)
15
5
0
-3
-4
-3
0
5
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Exercise
1. Draw the graph of the function y=x2-6x+5 find the least value of this function and the corresponding value of x
Solution
x
-3
-2
-1
0
1
2
3
4
5
y
32
21
12
5

0
-3
-4
-3
0
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Least value.
y= – 4 where x=3
2. Draw the graph of the function y =x2-4x+2 find the maximum function and the corresponding value of x use the curve to solve the following equations

a) X2-4x-2 = 0

b) X2-4x-2 = 3
b)
Solution
Table of values of y= x2-4x+2
x
-3
-2
-1
0
1
2
3
4
5
6
y
23

14
7
2
-1
2
-1
2
7
14
Find the maximum value of the function.;
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Maximum value
=EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Y= ( EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION ) = – 2
Find the maximum value of the function;
Maximum value= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
Maximum value= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
Maximum value
[2, -2]
The maximum value is = (-2,-2)

a) x2-4x-2 = 0
add 4 both sides

x2-4x + 2 = 4
but x2-4x+2 = y
... y = 4
Draw a line y = 4 to the graph above. The solution from the graph is

x1 = -1/2 , x2 = 9/2
(x1,x2) = (-1/2,9/2 )

(b) x2-4x – 2 = 3
add 4 both sides
x2-4x – 2 +4 = 3+ 4
x2-4x + 2 = 7

but x2-4x+2 = y
... y = 7
Draw a line y = 7 to the graph above. The solution from the graph is

x1 = -1 , x2 = 5
(x1,x2) = (-1,5 )
3. In the problem 3 to 5 write the function in the form f(x)= a( x + b)2 +c where a, b, c are constants
f(x) =5-x-9x2

Solution
f(x) = -9x2 – x + 5
= -9( x2 EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION ) + 5
= -9( x2EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION+ EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION) + 5 + EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
= -9(x – EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION)2 + EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
4. In the following functions find:
a) The maximum value
b) The axis of the symmetry
f(x)= x2– 8x+18
Solution
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION ) = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
( 4, 2)
Maximum value =2 where the axis of symmetry x=4.

5. f(x) = 2x2+3x+1

Solution
Maximum value EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
The turning point of the graph is EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION )
The minimum value of the graph is y = – EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
axis of symmetry = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
POLYNOMIAL FUNCTIONS
The polynomial functions are the functions of the form, ” P(x) =an xn + a n-1 xn-1 + an-2 xn-2 + . . . + a1 x1+a0 x0 “. Where n is non negative integer and an, an-1, an-2 . . . a0 are real numbers. The degree of a polynomial function is the highest power of that polynomial function.
Example
a) f(x) =5x4 -7x3 +8x2– 2x+3 is a degree of 4
b) H(x)=6x-8x2+9x9-6 is a degree of 9
c) G(x) =16x-7 is a degree of 1
d) M(x) =6 degree is 0 =6x0
GRAPHS OF POLYNOMIAL FUNCTIONS

EXAMPLE
Draw the graph of f(x) = x3-2x2-5x+6

Solution
Table of values
x
-3
-2
-1
0
1
2
3
4
F[x]
-24
0
8
6
0
-4
0
18
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
STEP FUNCTIONS

EXAMPLE
1. If f is a function such that;
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Draw the graph and find its domain and range.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Domain = {x: xEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTIONR, except -3< x < -2}
Range = {1,4, -2}
2. The function is defined by
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
a)
Sketch the graph of f(x) use the graph to determine the range and the domain. Find the value of f(-6) , f(0). State if it is a one to one function
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Domain
= {x : XEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTIONR}
Range
{ y:y >1}
f(-6) = -2
f(0)= 2
It is not a one to one function.

EXERCISE
1. Draw the graph of the following defined as indicated
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Solution:
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Solution
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION


2. Given that f(x) = EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
a) On the same set of axes sketch the graphs of f(x )and the inverse of f(x), From your graphs in [a] above determine;
(a)The domain and range of f(x)
(b)The domain and range if the inverse of f(x)
(c)Find f(- 5)and f(5)
(d)Is f (x)a one to one?
(e)Is the inverse of f(x) a function?
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
(a) Domain of f(x ) = { x: x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION }
(b) •Range of f(x) = { y : y EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION}
•Domain of f-1(x ) = { x : x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION}
(c) Range of f-1(x ) = { y: x EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION }
(d) f(-5) = 1 and f(5) = 6
(e)Yes the inverse of f(x) is a function and f(x) is one-to-one function
ABSOLUTE VALUE FUNCTIONS
The absolute value function is defined by f(x) EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION

Example

1. Draw the graph of f(x) = | x |
Solution
Table of values f(x) = | x |
X
-3
-2
-1
0
1
2
3
f(x)
3
2
1
0
1
2
3
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
THE INVERSE OF A FUNCTION
Given a functon y= f(x), the inverse of f(x) is denoted as f-1(x). The inverse of a function can be obtained by interchanging y with x (interchanging variables) and then make y the subject of the formula.
Example
1. Find the inverse of f(x) = 2x+3
Solution
Y= 2x+3
X=2y+3
2y=x-3
Y =EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
f-1(xEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
EXPONENTIAL FUNCTIONS
An exponential function is the function of the form f(x) = nx where n is called base and x is called exponent.
Example
1. Draw the graph of f(x)=2x
Solution :
Table of values
X
-3
-2
-1
0
1

2
3
f(x)
1/8
1/4
½
1
2
4
8
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
2. Draw the graph of f (x)=2-x

Solution:
Table of values
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Properties of exponential function
  • When x increases without bound, the function values increase without bound
  • When x decreases, the function values decreases toward zero
  • The graph of any exponential function passes through the point (0,1).
  • The domain of the exponential function consists of all real numbers whereas the range consist of all positive values.

LOGARITHMIC FUNCTION
Logarithmic function is any function of the form f(x)= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTIONread as function of logarithm x under base a or f(x) is the logarithm x base a
Example
Draw the graph of f(x)=EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Solution
Table of values

x
1/8
¼
1/2
1
2
4
8
f(x)
-3
-2
-1
0
1
2

3
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
THE INVERSE OF EXPONENTIAL AND LOGARITHMIC FUNCTION
The inverse of the exponential function is the relation of logarithmic in the line y=x
EXAMPLE
1. Draw the graph of the inverse of f(x) = 2x and f(x)= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTIONunder the same graph.

Solution
(i) y= 2x
Apply log on both sides
log x=log 2y
log x =y log 2
y = log (x- 2)
f-1(x) = log (x- 2)
(ii) f(x)=EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
y= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION

x= EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
y= 2x
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION


Table of values
X
1/8
1/4
1/2

1
2
4
8
f(x)
-3
-2
-1
0
1
2
3
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
2. Draw the graph of the function f(x)=3x
Table of values
x
-3
-2
-1
0
1
2
3
f(X)
1/27
1/9
1/3
1
3
9
27
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
3. Draw the graph of the function f(X)=8x
Solution
Table of values
X
-3
-2
-1
0
1
2
3
f(X)
1/512
1/64

1/8
1
8
64
512
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION
Exercise
1. Find the graph of y =2x and given that ¾ =2-0.42 draw the graph of f(x)= (3/4)x
Table of values if f(x) =(3/4)x
X
-3
-2
-1
0
1
2
3
2x
1/8
1/4
½
1
2
4
8
-0.42x
1.26

0.84
0.42
0
-0.42
-0.84
1.26
2-0.42x
64/27
16/9
4/3
1
3/4
9/16
27/64
Copy and complete the following table and hence draw the graph of f(x)=(1/4)x
X
-3
-2
-1
0
1
2
3
2x
-6
-4
-2
0
2
4

ecolebooks.com
6
-2x
6
4
2
0
-2
-4
-6
2-2x
64
16
4
0
1/4
1/16
1/64
[1/4]x
64
16
4
0
1/4
2/4
¾

EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - FUNCTION




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