2 function - composite-inverse

7/28/2019 2 Function - Composite-Inverse

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MAT133 4/14/20

2 Functions2 Functions2 Functions2 FunctionsObjectives:

To be able to identify the notation of compositefunctionTo be able to find the inverse of the function

HASFAZILAH AHMATJAN 2010

To find domain and range for composite and inversefunctions

CompositeComposite

For functions f and the com osite functionFor functions f and the com osite functiong by f, denoted has range values defined byg by f, denoted has range values defined by

( )( ) ( )( ) ( )= =Df g x f g x fg x

( )x ( )g x ( )( )f g x



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Example 1Example 1Which one is the same asWhich one is the same as

1.1. ff (x)(x)

( )( )Df g x

2.2. (fg)(x)(fg)(x)

3.3. f(x)g(x)f(x)g(x)

4.4. gf(x)gf(x)


Answer NowAnswer Now

Example 2Example 2Given andGiven and . Find the. Find thefollowingfollowing

a)a) f (x)f (x) b)b) f( (3))f( (3))

( ) = 3f x x ( ) = + 2g x 1 x

c)c) g(f(x))g(f(x)) d)d) gf(3)gf(3)

SolutionSolution

a)a) b)b)( )( ) ( )= + 2f g x f 1 x ( )( ) ( )= +3

2f g 3 1 3


( )( ) ( )= + 2f g x 1 x ( ) ( )=g 3 10( )( ) =f g 3 1000


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MAT133 4/14/20

Example 2 (cont.)Example 2 (cont.)

Given andGiven and . Find the. Find thefollowingfollowing

c)c) (f(x))(f(x)) d)d) f(3)f(3)

( ) = 3f x x ( ) = + 2g x 1 x

SolutionSolution

c)c) d)d)( )( ) ( )= 3g f x g x( )( ) ( )= + 3 2g f x 1 x

( )( ) ( )= 3g f 3 g 3( )( ) ( )=g f 3 g 27


( )( ) = + 6g f x 1 x ( )( ) ( )= +2

g f 3 1 27

( )( ) =g f 3 730

Example 3Example 3Given .Given . . Find the function of g. Find the function of gifif fgfg(x) is given by(x) is given by

( ) ( )= +2f x x 1 4( ) = +2fg x 4 x 20 x 29

SolutionSolution

( ) ( )( )2 2

fg x g x 1 4 4 x 20 x 29= + = +

( )( )2 2

g x 1 4 4 x 20 x 29 + = +

( )( )2 2

g x 1 4 x 20 x 29 4 = +


( )( )2 2g x 1 4 x 20 x 25 = +

( )( ) ( )2 2

g x 1 2 x 5 =

( )g x 1 2 x 5 = ( )g x 2 x 4=


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MAT133 4/14/20

Example 4Example 4

GivenGiven . Find the function of f if. Find the function of f iffhfh(x) is given by(x) is given by

( )h x x 5= +( ) 1fh x

x 2=

SolutionSolutionLetLet h( x ) y=

y x 5= +

( )1

f y =

x y 5=


y

( )1

f y y 7= Therefore, ( ) 1f x x 7=

Example 5Example 5GivenGiven and .and .Find , , and their domain and rangeFind , , and their domain and range

( )f x 3 x 2= +f gD

( ) 1g xx 3

=+

g fD ( )g g fD D

SolutionSolutionDomain of f(x) : all real numberDomain of f(x) : all real number Range : all real numberRange : all real number

Domain of g(x) : x Domain of g(x) : x --33 Range : g(x) 0Range : g(x) 0

1f g f

x 3

= +

D


( )fg x 3 2x 3= + +

( )9 2 x

fg xx 3

+=

+

Domain : x Domain : x --33Range :Range : fgfg(x) 2(x) 2


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MAT133 4/14/20

Example 5 (cont.)Example 5 (cont.)GivenGiven and .and .

Find , , and their domain and rangeFind , , and their domain and range( )f x 3 x 2= +

f gD

( )1

g x

x 3

=

+g fD ( )g g fD D



( )g f g 3 x 2= +D1

x =


3 x 2 3+ +

( ) 1gf x3 x 5

=+

Domain : x Domain : x --5/35/3Range :Range : gfgf(x) 0(x) 0

Example 5 (cont.)Example 5 (cont.)GivenGiven and .and .Find , , and their domain and rangeFind , , and their domain and range

( )f x 3 x 2= +f gD

( ) 1g xx 3

=+

g fD ( )g g fD D



1g g f g

3 x 5

= +

D D

1( )( )

3 x 5g gf x

16 3 x

+=

+


g x 13

3 x 5

= + + Domain : x Domain : x --5/3,5/3, -- 16/316/3Range :Range : gfgf(x) 1(x) 1

( )( )1

g gf x1 3( x 5 )

3 x 5

=+ +

+


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InverseInverse

Two functions f and g (oneTwo functions f and g (one--toto--one) are said to beone) are said to beinverse functions if and only ifinverse functions if and only if

1. For each x in the domain of x is in the1. For each x in the domain of x is in thedomain of f anddomain of f and

2.2. For each x in the domain of f, f(x) is in theFor each x in the domain of f, f(x) is in thedomain of g anddomain of g and

( )( ) ( )( )f g x f g x x= =D

( )( ) ( )( )g f x g f x x= =D


( )x ( )f x

f

1f

Domain fDomain f Range fRange f

Domain fDomain f--11Range fRange f--11

InverseInverse( )f x

6

7

8

9

y

( )1f x

7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 0 11

3

2

1

1

2

3

4

5

x


8

7

6

5

4

The graph of inverse is symmetrical to the line y = xThe graph of inverse is symmetrical to the line y = x


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MAT133 4/14/20

Example 1Example 1

GivenGiven andand ..Determine whetherDetermine whether

( ) 3f x x 1= ( ) 3g x x 1= +

( ) ( )1f x g x=

SolutionSolution

LetLet

Using the definition of inverseUsing the definition of inverse( )1g x y =

( )( )1g g x x =

( ) 3g y y 1 x= + =


3y 1 x+ =

3y x 1=

Therefore,Therefore,

( ) ( )1 3g x x 1 f x = =

Example 2Example 2Given h(x) = 5. Decide whether h(x) has an inverseGiven h(x) = 5. Decide whether h(x) has an inverseor not.or not.

, fin

x

yo u ono u on


Since h(x) is not a oneSince h(x) is not a one--toto--one function,one function,therefore, it does not have an inversetherefore, it does not have an inverse


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Example 3Example 3Let f be the function andLet f be the function and

findfind andand

( )2 x 4

f x

3

+= ( ) 2g x x 1=

( )1gf x( )1f x

SolutionSolution

LetLet

Using the definition of inverseUsing the definition of inverse ( )( )1f f x x =( )

2 y 4f y x

3

+= =

( )1f x y =


2 y 4 3 x+ =

3 x 4y2

=


( )1 3 x 4f x2

=

Example 3 (cont.)Example 3 (cont.)Let f be the function andLet f be the function andfindfind andand

( ) 2 x 4f x3

+= ( ) 2g x x 1= ( )1gf x( )1f x

SolutionSolution

( )( )1 3 x 4g f x g2

=

23 x 4

12

=


29 x 24 x 16 4

4

+ =

29 x 24 x 12

4

+= ( )( )

13 x 2 3 x 6

4=


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Example 4Example 4Sketch the following graph and its inverse on theSketch the following graph and its inverse on the

same coordinate. Determine the domain of thesame coordinate. Determine the domain of theinverse function.inverse function.

( ) = 3f x x 1

SolutionSolution

x

y

x

y

x

y

( )1f x


Example 5Example 5Sketch the following graph and its inverse on theSketch the following graph and its inverse on thesame coordinate. Determine the domain of thesame coordinate. Determine the domain of theinverse function.inverse function.

( ) = + f x x 1 , x 1

SolutionSolution=y x

( )f x

( )1f x



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Example 6Example 6Sketch the following graph and its inverse on theSketch the following graph and its inverse on the

same coordinate. Determine the domain of thesame coordinate. Determine the domain of theinverse function.inverse function. ( ) = + 2f x x 2, x 0

=y x

SolutionSolution

( )1f x

x


Example 7Example 7Find the inverse function of the following functions.Find the inverse function of the following functions.Determine the domain of each inverse.Determine the domain of each inverse.

( ) = x

f x , x 2

SolutionSolution

( ) =1f x y

( ) =f y xBy definition,By definition,( ) = =

+

yf y x

y 2

( ) =y 1 x 2 x

=

2 xy

1 x


( )= +y x y 2= +y xy 2 x

=y xy 2 x


( ) =

1 2 xf x1 x


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Example 7 (cont.)Example 7 (cont.)Find the inverse function of the following functions.Find the inverse function of the following functions.

Determine the domain of each inverse.Determine the domain of each inverse.

( ) = x

f x , x 2

SolutionSolution

So, the domain ofSo, the domain of

Domain : x 1Domain : x 1

( ) =

1 2 xf x1 x

Example 8Example 8Find the inverse function of the following functions.Find the inverse function of the following functions.Determine the domain of each inverse.Determine the domain of each inverse.

( )+

= 2 x 3

f x , x 1

SolutionSolution

( ) =1f x y

( ) =f y xBy definition,By definition,

( )+

= =

2 y 3f y x

1

( ) = y 2 x x 3

=

x 3y

2 x


( )+ = 2 y 3 x y 1

= 2 y xy x 3

= 2 y xy x 3


( )

=

1 x 3f x

2 x


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Example 8 (cont.)Example 8 (cont.)Find the inverse function of the following functions.Find the inverse function of the following functions.

Determine the domain of each inverse.Determine the domain of each inverse.

( )+

= 2 x 3

f x , x 1

SolutionSolution

So, the domain ofSo, the domain of

Domain : x 2Domain : x 2

( )

=

1 x 3f x2 x

2 function - composite-inverse

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