1 TOPIC 5: FUNCTIONS & GRAPHS

LECTURE 1 OF 7

TOPIC : FUNCTIONS & GRAPHS

SUBTOPIC : 5.1 FUNCTIONS

LEARNING OUTCOMES :

At the end of the lesson, students are able to: a) represent relation using arrow diagram or as ordered pairs. b) identify relations and functions. c) sketch graph function.

relations!

The correspondence between elements in one set and the elements in another set. For example

Let A={2,3} and B={6,15,12}. Relation: “ is a factor of ”.

Displayed by using :

Arrow diagram

Ordered pairs {(3,6),(3,12), ),(3,15), (4,12),(5,10)}.

A B

3

4

5

6

15

12

DEFINITION FUNCTION

A function is a relation in which every element of set X is matched with only one element of set Y.

2 TOPIC 5: FUNCTIONS & GRAPHS

Types of relation

One to one One to many Many to one Many to many

Function Not Function Function Not Function

DOMAIN CODOMAIN

A function f: A →B is a mapping from set A to set B, written as

f: x →y, or f(x) = y with and x is the object and y is the image.

Domain = set of all objects i.e. Set A

Range= set of all images .

Codomain = Set B. Each element in set A is mapped onto its image in set B but

may be some elements in Set B are mapped onto.

A function is onto when its range equals its codomain.

• A

• B

• C

• 1

• 2

• 3

• 1

• 2

RANGE

B A

a

b c

1 2 3

B A

a

b

c

1 2

3 3

1 2 b

a

c

B A

4

1

2

3

a

b

c

A B

3 TOPIC 5: FUNCTIONS & GRAPHS

Example 2: Illustrate the function 3: +→ xxf

Example 3:

The function f is given by 63: −→ xxf . Find

a) f(4)

b) the value of x whose image is 18.

c) the value of x such that f(x)=x.

f B A

1

2

3

4

Finding the value of image requires substitution of image into the function.

4 TOPIC 5: FUNCTIONS & GRAPHS

. a y x= 2. b y x=

Ans:

Ans:

( ) ( )2 . 2 – f x xc x= 1. d yx

= −

Ans:

Ans:

The Test of a function

VERTICAL LINE TEST

The vertical line test is a graphical method used to determine whether a relation in x is a function. If any vertical line drawn intersects the curve y = f(x) only at one point, then f(x) is a function of x.

EXAMPLE 1: State whether the graph represent a function or not?

f(x

x

f(x

x

f(x)

x

f

0 2

5 TOPIC 5: FUNCTIONS & GRAPHS

LECTURE 2 OF 7

SUBTOPIC : 5.1 FUNCTIONS

LEARNING OUTCOMES :

At the end of the lesson, students are able to:

a. Sketch the graph of a function. b. State the domain and range of a function.

Domain, 𝐷𝐷𝑓𝑓, is the set of the values of x in which 𝑓𝑓(𝑥𝑥) is defined. Some functions

defined for certain values of x only.

Range, 𝑅𝑅𝑓𝑓, is the set of all possible value of 𝑓𝑓(𝑥𝑥) as x varies throughout the domain .

Domain and range of function can be written in the form of sets or interval notations.

There are two methods to find the domain and range of a function 𝑓𝑓(𝑥𝑥)

(i) Graphically

(ii) Algebraically

The domain and the range of the function can be determined by means of graph,

How far the graph extents horizontally will determine the domain, and vertically to

determine the range.

DoMain & ranGe

6 TOPIC 5: FUNCTIONS & GRAPHS

For function f to be defined (or exists) ,we consider the kind of numbers that can be substituted into )(xf ; usually the negative values, the positives or zero. It is also said that domain is the set of restricted values that can be substituted so that f is defined.

TYPES OF FUNCTION RESTRICTION FOR THE DOMAIN

DESCRIPTION

1.Fraction g( x )f ( x )h( x )

0)( ≠xh It can be positive or negative only

2. Square root

f ( x ) g( x ) 0)( ≥xg

Value of number under a square root can be positive or zero only

3. Logarithmic

f ( x ) ln g( x ) 0g( x ) Take log of positive value only. it cannot be negative or zero.

4. Polynomials: constant, linear, quadratic and cubic functions

2 3f ( x ) x x

with Rx∈

All real numbers can be raised to the first, second and third power.

fD R

5.Exponential

xf ( x ) e Rx∈

All real numbers can be indices.

fD R

7 TOPIC 5: FUNCTIONS & GRAPHS

Determine the domain and range of each of the following functions:

a. 2( ) 4 3f x x x= − + b. 2( )4 3xf xx−

=+

8 TOPIC 5: FUNCTIONS & GRAPHS

i) Constant

f fD R

f fD R

ii) linear

f fD R

f fD R

(iii ) Quadratic function ( find vertex using Completing the square) (a) 0,)()( 2 >+−= akhxaxf

Open up

f fD R

(b) 2 0f ( x ) a( x h ) k ,a

f fD R

Basic shape graph of a function

x

k

x = k x

y

x

y

y = k

f(x)

x

k

x

y y = mx+c with m >0 y =mx+c

with m<0

x

y

9 TOPIC 5: FUNCTIONS & GRAPHS

(iv) Cubic function 3f ( x ) x c

f fD R

3f ( x ) x c

f fD R

(v) surd function

(a) f ( x ) x c d

f fD R

(b) f ( x ) x c d

f fD R

(vi) Reciprocal function

f fD R

f fD R

1f ( x )x a

x

xf 1)( −=

f(x)

0

f(x)

0 c

f(x)

0 x

x

x

f(x)

d

f(x)

d

c x

10 TOPIC 5: FUNCTIONS & GRAPHS

vii)Absolute value function a . f(x) = |x-c|+d

f fD R

b. f(x) = -|x-c|+d

f fD R

(viii)Piecewise function Combine more than one functions together based on their respective domains

>+−=<

=1,1

1,51,

)(xx

xxx

xf

f(x)

0 c x d

f(x)

0 C x

11 TOPIC 5: FUNCTIONS & GRAPHS

Sketch the graph of the functions and state the domain and range for each of the functions.

1. y = 5 2. x=4

Rf fD = = Rf fD = =

2. y = 4x + 3 4. y = 2 - 2x

Rf fD = = Rf fD = =

12 TOPIC 5: FUNCTIONS & GRAPHS

2. Quadratic functions

The general form 2( )f x ax bx c= + + can be converted to this form ( )2( )f x a x h k= − + by using completing the square.

The minimum or maximum point of a quadratic function is V(h,k) or ,2 2b bfa a

− −

.

1. 23 1y x= +

2. 2 1y x= − −

1.

2.

3. ( )2( ) 3 4f x x= − − 4. 36)( 2 ++−= xxxg

Rf fD = = Rf fD = =

13 TOPIC 5: FUNCTIONS & GRAPHS

3. Cubic function

The general form is 3 2( )f x ax bx cx d= + + + , where a, b, c and d are constant; 0a ≠

1. 3f(x) = x 2. 3f(x) = -x

Rf fD = = Rf fD = =

5. xxxf 6)( 2 −= 6. 4122)( 2 +−= xxxf

Rf fD = = Rf fD = =

14 TOPIC 5: FUNCTIONS & GRAPHS

3. 3g(x) = 4 - x 4. ( )2 f(x)=2x 1x −

Rf fD = = Rf fD = =

4. Square root Functions or Surd Functions The domain of the square root function are the set of non – negative real numbers. The basic shape of surd functions:

( )f x x= − ( )f x x=

( )f x x= − − ( )f x x= − 4. Sketch the graph of the functions and state the domain and range for each of the functions.

1. ( ) 2f x x= − 2. ( ) 3f x x= +

Rf fD = = Rf fD = =

15 TOPIC 5: FUNCTIONS & GRAPHS

3. ( ) 4f x x= + 4. 3+−= xy

Rf fD = = Rf fD = =

5. ( ) 1 2g x x= + + 6. ( ) 1 3 1g x x= − −

Rf fD = = Rf fD = =

16 TOPIC 5: FUNCTIONS & GRAPHS

5. Reciprocal Functions The domain of a reciprocal functions are all real values except where the denominator

equals zero 0.

For ( ) af x cx b

= +−

, line x-b and y=c are the asymptotes.

1. 1( )g xx

= 2. 1( )g xx

= −

Rf fD = = Rf fD = =

3. 1( ) 3g xx

= − 4. 3( ) 2f xx

= −

Rf fD = = Rf fD = =

17 TOPIC 5: FUNCTIONS & GRAPHS

5. 4( )2

f xx

=+

+1 6. 5( )3

f xx−

=−

+2

Rf fD = = Rf fD = =

5. Absolute values of linear function

1. ( )g x x= 2. ( )g x x= −

Rf fD = = Rf fD = =

18 TOPIC 5: FUNCTIONS & GRAPHS

3. ( ) 4g x x= + 4. ( ) 4 1f x x= −

Rf fD = = Rf fD = = 5. ( ) 2 3f x x= − + − 6. ( ) 1 3 1f x x= − − +

Rf fD = = Rf fD = =

19 TOPIC 5: FUNCTIONS & GRAPHS

5. Piecewise Functions

Functions that are defined by more than one expression for different intervals of the domain are called piecewise functions.

1.

>≤−

=3;2

3;)(

2

xxxx

xf

2.

>−≤<−

−≤=

4;441;

1;3)(

xxxx

xxf

Rf fD = = Rf fD = =

20 TOPIC 5: FUNCTIONS & GRAPHS

3. Write ( ) 1 5 5f x x x= − + as a piecewise function. Hence sketch the graph of the function. Write down its domain and range.

21 TOPIC 5: FUNCTIONS & GRAPHS

LECTURE 3 OF 7

TOPIC : FUNCTIONS & GRAPHS

SUBTOPIC : 5.2 COMPOSITE FUNCTIONS

LEARNING OUTCOMES :

At the end of the lesson, students are able to: a) Represent composite function by an arrow diagram. b) Find composite functions. c) Find one of the functions when the composite and the other functions are given.

Composite functions are functions that arise as a result of the combination of two or more basic functions.

g f is denoted by ( )( ) [ ]( )g f x g f x= .

f g

g f Or gf

•x •f(x) •g [f(x)]

CoMposite funCtions

22 TOPIC 5: FUNCTIONS & GRAPHS

a. fg(x)

b. gf(x) c. gg(x)

a. gf(3)

b. fgh(2) c. ghf(4)

EXAMPLE 1

If ( ) , 0 and g(x) = 1+ x, xf x x x= ≥ ∈ℜ . Find: a. fg(x) b. gf(x) c. gg(x)

EXAMPLE 2

2

( ) 3 , , g(x) = 2x - 1, x and h(x) = x ,If f x x x R R

x R= ∈ ∈

∈. Find:

a. gf(3) b. fgh(2) c. ghf(4)

23 TOPIC 5: FUNCTIONS & GRAPHS

Solution:

The functions f, g and h are defined by ( ) 2f x x= − , 3( ) , 11

g x xx

= ≠ −+

and

( ) 2 1h x x= − .

(a) Show that ( )2f x x= .

(b) Find an expression for ( )2g x , and state for which two values of x it is undefined.

(c) Solve the equation ( )3h x x= .

EXAMPLE 3

24 TOPIC 5: FUNCTIONS & GRAPHS

Solution:

Solution:

EXAMPLE 4 The function g is defined by 1,1)( −≥+= xxxg . Determine the function f (x) if ( ) xexgf =

Given that 𝑔𝑔(𝑥𝑥) = 𝑥𝑥2 + 1 and 𝑔𝑔𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 4𝑥𝑥 + 5, find the function of 𝑓𝑓(𝑥𝑥).

EXAMPLE 5

25 TOPIC 5: FUNCTIONS & GRAPHS

Solution:

EXAMPLE 6 The function f is defined by ( ) 2, x R.f x x= − ∈ . Determine the function g if 2( ) 6 1,gf x x x x R= − + ∈

EXAMPLE 7 If 2( ) 4 2 1fg x x x= − + and ( ) 2 1g x x= + , find the function of gf(x). Subsequently, find the values of x that satisfy fg(x) = gf(x).

In both diagrams, every element is mapped onto exactly one element of set B.

In diagram 1 function f has the property of one-to-one function.

If f(x)=3, then x=a only

: However in Diagram 2, elements a and c are mapped onto element 1, which is many to one.

Since cabutcgag ≠= )()( , therefore g is not one to one.

LECTURE 4 AND 5 OF 7 TOPIC : FUNCTIONS & GRAPHS SUBTOPIC : 5.3 INVERSE FUNCTIONS LEARNING OUTCOMES :

At the end of the lesson, students are able to: a) To determine whether a function is one to one. b) Define and determine the inverse of a function. c) Determine whether the function has an inverse and find the inverse of a function d) Identify the domain and range of and inverse function

ONE-TO-ONE FUNCTIONS

B A

a

b

c

1 2

3

B A

a

b c

1 2 3

Diagram 1 Diagram 2

f g

To show f is one to one: 2121 )()( xxthenxfxfIf ==

26

Example 1: Use the graphical method to determine whether each of the following

functions is one-to-one function.

a)

b)

c)

d)

-1

1

x

f(x)

-

1 x

f(x)

-1

1

x

f(x)

1

METHODS TO DETERMINE ONE-TO-ONE : -Graphical and Algebraic Methods

Grap

hica

l m

etho

d

• Horizontal Line test If any horizontal line drawn intersects the graph of the function cuts only

at one point, then the function is one-to-one function. It shows that one

image is mapped from only one object.

0 2 x

f(x

27

Example 2: Determine whether f is a one to one function or not . a/ 2 3f x x

(b) 3 12f x x

c) 2 6f x x x d) 2 12 6f x x x; x

2121 )()( xxthenxfxfIf ==

• Al

gebr

aic

Met

hod

Function f maps elements Set X to those in Set Y. A function f is called a one-to-one function if two elements of set X have the same image. Note: Quadratic or absolute value functions are symmetrical and not one to

one. To make it one-to-one, we have to take either side of the graph.

28

If f maps the elements from set X to Set Y. The inverse function of f or 1f x will map back from Y to X . X Y

Note that

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝑜𝑜 𝑜𝑜−1 = 𝑅𝑅𝐷𝐷𝐷𝐷𝑅𝑅𝑅𝑅 𝐷𝐷𝑜𝑜 𝑜𝑜 = 𝑆𝑆𝑅𝑅𝑆𝑆 𝑋𝑋 𝑅𝑅𝐷𝐷𝐷𝐷𝑅𝑅𝑅𝑅 𝐷𝐷𝑜𝑜 𝑜𝑜−1 = 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝑜𝑜 𝑜𝑜 = 𝑆𝑆𝑅𝑅𝑆𝑆 𝑌𝑌

To verify that 1f x is the inverse of f, show that xxff =− )(1

•

Existence of The Inverse of a Function

𝐼𝐼𝑜𝑜 𝑜𝑜−1𝐷𝐷𝑖𝑖 𝐷𝐷 𝐷𝐷𝐷𝐷𝑅𝑅 − 𝑆𝑆𝐷𝐷 − 𝐷𝐷𝐷𝐷𝑅𝑅 𝑜𝑜𝑓𝑓𝐷𝐷𝑓𝑓𝑆𝑆𝐷𝐷𝐷𝐷𝐷𝐷, 𝑆𝑆ℎ𝑅𝑅𝐷𝐷 𝑜𝑜−1𝑅𝑅𝑒𝑒𝐷𝐷𝑖𝑖𝑆𝑆𝑖𝑖

And:

𝐷𝐷𝑓𝑓−1 = 𝑅𝑅𝑓𝑓 𝑅𝑅𝑓𝑓−1 = 𝐷𝐷𝑓𝑓

How? Step 1: Let y=f(x), then f-1(y)=x

Step 2: Rearrange so that x becomes the subject of equations.

Step 3: x is the inverse function as stated in step 1.

Or

Use property xxff =− )(1

-3

0

1

-11

-2

1

f

f-1 X Y

1

Let y f ( x ),then x f ( y )

INVERSE FUNCTIONS

29

Example :Find the inverse function of

a) 24

f ( x )x

b) 2 4 4 2f ( x ) x x ; x .

2. Given f(x) = 3−

+x

px , 3≠x , where p is a constant. Determine

a) the value of p if f(5) = 23 ,

b) f -1

30

c) the value of x for which f -1 is undefined

3. Given that 43)( 3 += xxf and 2

3)(−

=x

xxg

a) State the domain and range for f and g b) Find f-1 and g-1 and the domain and range

31

4. The functions f and g are defined by 352:

+−

→xxxf and

xxxg−+

→2

53: .

a) Find )(xfg and deduce )(1 xf −

b) Determine the domain and range of ( )xf 1−

32

LECTURE 6 AND 7 OF 7 TOPIC : FUNCTIONS & GRAPHS SUBTOPIC : 5.4 EXPONENTIAL AND LOGARITHMIC FUNCTION LEARNING OUTCOMES :

At the end of the lesson, students are able to: a) Determine the relation of exponential and logarithmic functions by algebraic and graphical approaches. b) Find the domain and range of an exponential and logarithmic functions c) Determine the composite functions involving exponential and logarithmic function. d) Sketch the graph involving exponential and logarithmic functions

Exponential and Logarithmic Function

Relationship Between An Exponential and Logarithmic Function Exponential function The inverse of exponential function, 𝑜𝑜(𝑒𝑒) = 𝐷𝐷𝑥𝑥 is Let 𝑜𝑜[𝑜𝑜−1(𝑒𝑒)] = 𝑒𝑒

xa )x(f =−1

xlog)x(f a=−1 The inverse of exponential function is a logarithmic function. Logarithmic function The inverse of logarithmic function, 𝑜𝑜(𝑒𝑒) = log𝑎𝑎 𝑒𝑒 is

Let 𝑜𝑜[𝑜𝑜−1(𝑒𝑒)] = 𝑒𝑒 log𝑎𝑎 𝑜𝑜−1(𝑒𝑒) = 𝑒𝑒 𝑜𝑜−1(𝑒𝑒) = 𝐷𝐷𝑥𝑥

The inverse of logarithmic function is exponential function.

y = x

f(x)=logax

f (x) = ax 1

y

x

33

a)For f(x) = ax +k ,where x ∈ R and a

+→−∞→

∞→∞→

kyxAsandyxAs

,,

b) For f(x) = a-x +k ,where x ∈ R and a >1

+→∞→

∞→−∞→

kyxAsandyxAs

,,

c) For f(x) = -ax +k ,where x ∈ R and a >1

−→−∞→

−∞→∞→

kyxAsandyxAs

,,

a) For f(x) = -a-x +k ,where x ∈ R and a >1

−∞→−∞→→∞→ −

yxAsandkyxAs

,,

Sketch the graph of the following functions:

a) 2 1xf ( x )

EXAMPLE 1

Basic Exponential Function Graph Domain is R

• Horizontal asymptote : line y=k

• Takes a bend shaped curve and cuts y -axis.

• The one end of the curve will approach the asymptotic line and the other end will

increase or decrease infinitely.

Exponential Function:

Exponential Function is a function of the form f(x) = ax where x ∈ R and a > 0

To deduce…

34

b)

23 1xh( x )

c) 2xh( x ) e d) 5xy e

35

e) 12 −−= −xey Function of the form ( )kxxf a −= log)( , x ∈ R+, a > 0

a) Domain is x > k (tips: +x for RHS of VA)

b) Vertical asymptote ; line x=k

c) Take a bend shaped curve and always cut x –axis

d) One end of the curve will approach the asymptote,

and another one will increase/decrease infinitely

For ( )kxxf a −= log)(

−∞→→

∞→∞→

∞=

+ ykxAsandyxAs

kD f

,,),(

( )xkxf a −= log)(

−∞→→

∞→−∞→

−∞=

− ykxAsandyxAs

kD f

,,),(

For ( )kxxf a −−= log)(

∞→→

−∞→∞→

∞=

+ ykxAsandyxAs

kD f

,,),(

( )xkxf a −−= log)(

∞→→

−∞→−∞→

−∞=

− ykxAsandyxAs

kD f

,,),(

Logarithmic Functions.

36

a) ( )1log)( 10 −= xxf b) ( )62ln)( −= xxf c) ( )3ln)( −= xxf

EXAMPLE 1 Sketch the graph of the following functions

37

d) ( )2ln)( −−= xxf e) ( )1ln)( +−= xxf f) )4ln()( xxf −−=

38

Relationship Between An Exponential and Logarithmic Function

Example 1 Given 2)( += xexf , find )(xf 1− and sketch the graph of )(xf and

)(xf 1− on the same axes.

39

Example 2 Given )23ln()( += xxf , determine )(xf 1− and sketch the graph of )(xf and

)(xf 1− on the same axes.

40

Example 3 The function f and g are defined as f(x) = ln (x + 1) , x > -1 and g(x) = x2 + 2x, x > -1

a) State the range of f and find the inverse function of f. b) Find the composite function gf .

c) Find the value of x if 131 −=− e)x)(gf(

41