(chapter 1) Intruction To Function

QQM1023 Managerial Mathematics input

Function

2.1 INTRODUCTION Example 1: Ali invested RM100 at a bank and earns simple interest rate at an annual rate of 6%. The interest I (output) depends on how long that the money have been invested, t (input). Then it can be shown that interest and time are related by the formula:

I = 100 (0.06) t Where I is in RM and t is in years. For example; in the first year Ali will earns:

I = 100 (0.06)(1) = RM

6

from his investment.

First year: t=1

How much will Ali earns if he invested his money for 10 years??

As we can see, the interest I (output) depends on the length of time t (input) that Ali’s money is invested. To express this dependence, we say that I “is a function of” t. Functional relations like this are usually specified by a formula that shows what must be done to the input to find the output. We can think of formula as defining “rule”: multiply t by 100(0.06) The rule assigns to each input number t exactly one output number I, which we symbolize by the following arrow notation:

t Æ I Chapter 2 : Introduction To Function

or

t

Æ

100(0.06)t 44

output

QQM1023 Managerial Mathematics

2.1

INTRODUCTION TO FUNCTION

A function is a rule that assigns to each input number exactly one output number Independent variable - a variable that represents input numbers for a function . Eg : t in example 1 Dependent variable

- a variable that represents output numbers for a function. Eg: I in example 1

Example 2:

y = x + 2 . Determine whether y

Given

is/isn’t a function of

x?

Solution •

if

x =1

•

if

x = −4

Therefore

∴

The rules define/not define

Chapter 2 : Introduction To Function

y

as a function of

x

.

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QQM1023 Managerial Mathematics

Example 3

y2 = x

Consider:

Try to input

x = 9 into the given function

Then,

Therefore

∴

The rule define/ do not define

y

as a function of

x .?

2.1.1 FUNCTIONAL NOTATION Usually the letters

f , g , h, F , G

and so on are used to represent

function rules. For example, the equation

y = x + 2 defines y as a

function of x, suppose we let ƒ represent this rule ( add 2 to the input). Then we say ƒ is the function. To indicate ƒ assigns the output 3 to the input 1, we write

ƒ (1) = 3, which is read “ƒ of 1 equals 3”. Generally, if x is any input, we have the following notation:

ƒ (x) (read as : “ƒ of x”) means the output number that corresponds to the input number x.

Chapter 2 : Introduction To Function

46

QQM1023 Managerial Mathematics thus the output of ƒ (x) is the same as y. Therefore we may write the equation

y = x + 2 as: y= ƒ (x) = x + 2 or simply

ƒ (x) = x + 2

Example 4: a)

Given a function

f (x ) = x + 7 .

corresponds to the input x =1 or

What is the output

f (1) ?

Solution:

b)

Find the value

f (3)

for the function

f (x) = 2 x − 1

Solution:

c)

Given

h(u ) =

h(u − 4 )

u+4 u

, find

h(5), h(− 4) and

Solution:

Chapter 2 : Introduction To Function

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QQM1023 Managerial Mathematics

d)

 1  f ( x ) = 4 , find f (4 ), f   and 100   f ( x + 4)

Given

Solution:

2.2 & 2.3 DOMAIN AND RANGE For a function f defined by an expression with variable x, the implied domain of f is the set of all real numbers variable x can take such that the expression defining the function is real. Domain : is a set consist of all valid input for a function

The range of f is the set of all values that the function takes when x takes values in the domain. Range : is a set of all valid output for a function (produce by the values in the domain) •

There are two ways to determine the domain and the range of a function : a) from the graph sketches b) Using Algebra

Chapter 2 : Introduction To Function

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QQM1023 Managerial Mathematics a) Domain and range from the graph sketches.

Example 5:

Example 6 : y

y

4 x

x 2

Domain : { x Є R } Range : { y Є R}

Domain : {x >2} Range : {y >4}

Example 7:

Example 8 :

y y Å

5

range Æ

3

3 1

ÅdomainÆ 2

4

x

Domain = { 2 ≤ x ≤ 4 } Range = { 3 ≤ y ≤ 5 }

Chapter 2 : Introduction To Function

-1

x

Domain = { x ≤ -1 } Range = { y = 1, y > 3}

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QQM1023 Managerial Mathematics

Example 9 :

Example 10 :

y

y

6 4

4

2

2

1

0

Domain = { Range = {

}

x

3

2

0

x

8

4

Domain = { Range = {

}

}

}

2.2.1 & 2.3.1 TYPES OF FUNCTION AND ITS DOMAIN & RANGE A. Constant Function :

Example 10:

y=k

@

x=h

y =5

Example 11:

y

x = 50

y x = 50

5

y=5

0 Domain = { Range = {

x } }

Chapter 2 : Introduction To Function

0

50 Domain = { Range = {

x } }

50

QQM1023 Managerial Mathematics B. Linear Function :

y = mx + c

Example 12 : y = 2x + 4

Example 13 : y = -5x + 10

-2 Domain = { Range = {

y

y

4

10

0

x

} }

0

2

x

Domain = { Range = {

} }

C. Quadratic Function : y = ax2 + bx + c

Example 14 : y = x2 + 2x - 8

Example 15 : y = -x2 + 10x -21

y

y 44 3

-4

-1

22

5

7

x

x

-9 Domain = { Range = {

} }

Chapter 2 : Introduction To Function

Domain = { Range = {

} }

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QQM1023 Managerial Mathematics

D. Polynomial/Cubic Function

y = ax3 + bx 2 + cx + d

Example 16 : y = x3 + 7

Example 17 : y = -x3 - 5

y

y

7

x x -5

Domain = { Range = {

} }

Domain = { Range = {

} }

E. Absolute Function

Example 18: f(x) = | x-1| The function could be separated into: f(x) = |x-1| (x-1) ; x ≥1 = -(x-1) ; x<1 Graph: only consider the positive value for the output (y)

y f(x) = 1-x

f(x) = x-1

x 0 Domain = { Range = {

1

} }

Chapter 2 : Introduction To Function

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QQM1023 Managerial Mathematics

EXERCISE 1: Name each function and determine their domain: a)

f ( x ) = 7 (constant function) Answer:

b)

y = 3 x + 1 is …..

Domain for

y = x 2 − 9 is …..

f ( x ) = x3 + x − 1(polynomial / cubic function) Answer:

e)

Domain for

y = x 2 − 9 (quadratic function) Answer:

d)

f ( x ) = 7 is ……

y = 3 x + 1(linear function) Answer:

c)

Domain for

Domain for

f ( x ) = x3 + x − 1 is …..

f ( x) = 5 x − 1 Answer: Domain for

f ( x) = 5 x − 1

Chapter 2 : Introduction To Function

is ……

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QQM1023 Managerial Mathematics F . Composite Function

Example 19 :

Example 20 :

2; x≥0

x+3 f(x) = 1 x

f(x) = 1; x<0

; x>1 ; -1<x <1 ; x≤-1

y

y

4 2 1 1 x -1 x

1 -1

0

Domain = { Range = {

}

Domain = { Range = {

}

} }

Example 21 : f(x) =

x+3 1

; ;

x≠1 x=1

y Domain = { Range = {

4

} }

1 x 1

Chapter 2 : Introduction To Function

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QQM1023 Managerial Mathematics

EXERCISE 2 1. Determine the domain for each composite function:

a) f(x) =

for x = 3 4  2  x for 1 ≤ x < 3

Domain = {

b) h(x) =

3 for  − 3 for

Domain = {

}

x≥2 x<2 }

2. Given the function f(q),

 q, for − 1 ≤ q < 0  f (q ) = 3 − q, for 0 ≤ q < 3  2q 2 , for 3 ≤ q < 5  determine; 1 2

a) f( − ) b) f(0) c) f(2) d) f(3) e) f(4) f) Domain for the function f(q) Domain = { q : } Chapter 2 : Introduction To Function

55

QQM1023 Managerial Mathematics G. Rational Function :

f(x) = p(x) q(x)

Example 23: f(x) = 2 Æ p(x) - numerator x+1 Æ q(x) - denominator 2.1.2 JENIS-JENIS FUNGSI DAN GRAF There are two ways to determine the domain for a rational function 1 : From the graph sketches

y

1. Fungsi Malar 2. Fungsi Linear 3. Fungsi Kuadratik 4. Fungsi Polinomial Kubik -1

5. Fungsi Rasional

0

x

6. Fungsi Punca Kuasa Dua 7. Fungsi Rencam 8. Fungsi Mutlak Based on the graph, we can determine the domain and the range for the given function, Domain = { } Range = { } 2 : Using Algebra In order to produce an output for a rational function, the denominator cannot be a zero (we cannot divide by 0). Therefore, the domain of a rational function can take any real numbers as an input EXCEPT the one that make the denominator equal to 0.

1.

FUNGSI MALAR

Persamaan amnya ialah

f ( x) =

y=k

2 x +1

x + 1≠ 0

In order for us to determine the domain for a rational function, we need

Misalnya, to find the values of x that make the denominator equal to zero Æ these cannot be3an input ynumbers. y=

Thus we set the denominator NOT equal to 0 and solve for x , Bentuk graf, x+1≠ 0 y=3 x ≠ -1 Therefore we have, the domain for the function f is all real numbers } EXCEPT -1 OR can be written as {

Chapter 2 : Introduction To Function

56

QQM1023 Managerial Mathematics

EXERCISE 3: Determine the domain for each given function:

a)

x+2 f(x) = 2 x − 81

b) h(x) =

c) g(y) =

3x − 1 2x + 5

4 y ( y + 1)( y − 2)

Chapter 2 : Introduction To Function

57

QQM1023 Managerial Mathematics

H. Square Root Function :

y=

p (x)

or

y = ( p ( x))

1 2

Example 23 : y = + x 1) From Graph Sketches : y Domain = { Range = {

} }

x

0

Example 24 : f ( x) = 6 x + 3

6x + 3 ≥ 0

2) Using algebra

6x + 3

is a real number if

If 6 x + 3 is negative, then imaginary number).

6 x + 3 is greater or equal to 0. 6 x + 3 is not a real number. (It is an

Since function values must be real numbers, we must assume that,

6 x + 3 ≥ 0, 6 x ≥ −3 x≥

−1 2

Thus the domain for f is greater or equal to –

{

Chapter 2 : Introduction To Function

½ or can be written as

}

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QQM1023 Managerial Mathematics

EXERCISE 4: Determine the domain for each given function:

a) h(x) =

x−3

b) g(x) =

4x + 3

6 c) f(x) =

2x − 8

Chapter 2 : Introduction To Function

59