composition-2.docx

7/26/2019 composition-2.docx

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Worksheet 2: Composition of functions

Name: Date:

Let there be two functions dened as:

f:AB by f(x) for allxA

g: BC by g(x) for allxB

Example 1:

Let two functions be defined as:

f={(1,2) ,(2,3) ,(3,4) ,(4,5) } and g={(2,4) ,(3,2) ,(4,3) ,(5,1) }

Check whethe !gof" and !fog" e#ist fo the gi$en functions%

Solution:

Domain Range &ence,

'ange of !f"oain of !g" !gof"e#ists%

'ange of !g"oain of !f" !fog" e#ists%

f {1,2,3,4} {2,3,4,5}

g {2,3,4,5} {4,2,3,1}={1,2,3,4}

*t eans that both co+ositions !gof" and !fog" e#ist fo the gi$en sets%

Then, the new function, gof!read as "g circle f" or "g

co#$osed with f", is denedas:

( )( ) ( ( ))g f x g f x=ofor all xA

%ange of g(x)%ange of h(x) &%ange of f(x)'o#ain of'o#ain off(x)


2/10

Example 2:

i$en ( ) 2 1f x x= + and2( ) 3g x x= + , find:

a% ( f o g)(#)%b% ( g o f)(#)%

c% ( f o f)(#)%

d% ( g o g)(#)%

Solution:

a. ( ) ( ) ( )( ) ( ) ( ) 22 22 1 2 -3 3g x xf g x f f xx= = = + + = + +o

.

b. ( ) ( ) ( )( ) ( ) ( )

22 31 2 1f x xg f x g g x= = + += +o

( )

2 234 4 1 4 4 2x x xx= + = + ++

c. ( ) ( ) ( )( ) ( ) ( )2 1 31 422 1f x xf f x f f xx= = = + =+ + +o

. ( ) ( ) ( )( ) ( ) ( )2

22 3 33g g x g gg x x x + = += = +o

( )4 2 4 2. / 3 . .x x x x= + + = +

0ote that:

( ) ( ) ( ) ( )2 22 - 4 4 2f g x x g f x x x= + = +o o

hat is, ( f o g)(#) is not the sae as (g o f )(#)% he o+en dot o is not the sae as auti+ication dot , no does it ean the sae thing%

f(#) g(#) = g(#) f(#) awa6s tue fo uti+ication7

%%%6ou cannot sa6 that:

( f o g)(#) = (g o f )(#) genea6 fase fo co+osition7

Domain an range of the composition of functions

Conside the function:

1( ) ,

1f x

x=

when 1x oain of fis { }18: Rxx , i%e% a ea nubes e#ce+t 1%


3/10

Let us now see the e#+ession of co+osition of function with itsef,

( ) ( ) ( )( ) 11

11 1

1

f xf x f fx

x

f = = =

o

1 1 1 1

1

1 1

1

1 x

x x

x x x x

x x

= = = =

$aid fo ea $aues of #9%

;ince f is undefined fo # = 1, and f fo is undefined fo # = , thus the doain of the co+osition

( ) ( )f f xois : { }1,8: xxRxx < i%e% a ea nubes e#ce+t and 1%

Sometimes !ou ha"e to be careful #ith the omain an range of the composite function.

$eneral rule to etermine the omain:

( )f x o6noia 1( )f x

x=

( )f x x= ( ) og( )f x x= ( ) xf x a=

Domain x x> x> , fo a ,{ }

,fo a=

Example:

i$en ( )f x x= and ( ) 3g x x= , find the doains of ( f o g)(#) and (g o f )(#)%

Solution:

( )f x x= x>;o:

( ) ( ) ( )( ) 3 3 3f g x f xg x xx= = > >o

&ence, the doain of ( f o g)(#) is a # > 3%

Now do the other composition:

( ) ( ) ( )( ) %%%gg x g xf = =o

&ence, the doain of (gof)(#) is ?

*


4/10

$oing back#ar: gi"en compose function% fin original functions

@sua6 co+osition is used to cobine two functions% Aut soeties 6ou ae asked to go backwads%

hat is, the6 wi gi$e 6ou a function, and the6B ask 6ou to coe u+ with the two oigina functions

that the6 co+osed%

Example 1:

i$en( ) ( ) ( )

25 3 5 -x xh x = + + +

, deteine two functions f (#) and g(#) which, when co+osed,

geneate h(#)%

Solution:

his is asking 6ou to notice +attens and to figue out what is inside soething ese%

*n this case, this ooks siia to the uadatic 2 3 -x x+ , e#ce+t that, instead of suaing #, the6Be

suaing # D 5%

;o etBs ake g(#) = # D 5, and then +ug this function into ( ) 2 3 -x x xf = +

:

( ) ( ) ( )( ) ( ) ( ) ( )2

5 35 5 -g x xf g f xf xx = = = ++ + + o

hen h(#) a6 be stated as the co+osition of ( ) 2 3 -x x xf = + and g(#) = # D 5%

Example 2:

i$en( ) 3 4h x x= +

, deteine two functions f (#) and g(#) which, when co+osed, geneate h(#)%

Solution:

;ince the suae oot is on (o aound) the 3# D 4, then the 3# D 4 is +ut inside the suae oot,

that is:

( ) ( ) ( ) ( )+

( ) ( ( ))

3 4 3 4

h x f g x

g x f xx x x

=

+ +

hus, g(#) = 3# D 4,( )f x x=

, and h(#) = ( f o g)(#)%


5/10

Exercise

Eo the gi$en functions:

a% f(#) = # D 1 , g(#) = 3#

b%2( ) 1, ( ) 2f x x g x x= =

c%2( ) 1, ( ) 5f x x g x x= =

d% f(#) = 2# D 1 , g(#) = #2

Eind:

1% oain and ange of each f(#) and g(#)

( )f x ( )g xa% oain =

'ange =

oain =

'ange =

b% oain =

'ange =

oain =

'ange =

b% oain =

'ange =

oain =

'ange =

c% oain =

'ange =

oain =

'ange =

2% eteine( ) ( )f g xo

and its doain

a% ( ) ( )f g xo

oain =

b% ( ) ( )f g xo

-


6/10

oain =

c% ( ) ( )f g xo

oain =

d% ( ) ( )f g xo

oain =

3% eteine( ) ( )g f xo

and its doain

a% ( ) ( )g f xo

oain =

b% ( ) ( )g f xo

oain =

c% ( ) ( )g f xo

oain =

d% ( ) ( )g f xo

oain =

4% eteine( ) ( )f f xo

and its doain

a% ( ) ( )f f xo

.


7/10

oain =

b% ( ) ( )f f xo

oain =

c% ( ) ( )f f xo

oain =

d% ( ) ( )f f xo

oain =

5% eteine( ) ( )g g xo

and its doain

a% ( ) ( )g g xo

oain =

b% ( ) ( )g g xo

oain =

c% ( ) ( )g g xo

oain =

d% ( ) ( )g g xo

oain =

/


8/10

.% F function is defined fo ea $aues b6 :

1( )

1

f x

x

=

fo a ea $aues e#ce+t # =1 %

eteine( )( )( )f f f x

and daw the ga+h of esuting co+ositionG

-% i$en f(2) = 3, g(3) = 2, f(3) = 4 and g(2) = 5, e$auate (f o g)(3)G

H% Eunctions f and g ae as sets of odeed +ais

f = {(I2,1),(,3),(4,5)} and g = {(1,1),(3,3),(-,/)}

Eind the co+osite function defined b6 g o f and descibe its doain and ange%

/% Jite function E gi$en beow as the co+osition of two functions f and g, whee

1( )g x

x=

and1

( )1

xF xx

=

+

0


9/10

1% K$auate f(g(h(1))), if +ossibe, gi$en that( ) , ( ) 1,h x x g x x= =

and

1( )

2f x

x=

+ %

11% Eo the co+osite function( )( )f g xo

and ( )f x , find( )g x

G

a% ( )( )f g x x=o

,2( ) 2f x x= +

b% ( ) . 2( ) 2 1f g x x x= + o

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

c%( ) ( )

2( ) 1 4f g x x= o

,( ) 2 4f x x=

d% ( ) ( ) 2( ) , 5f g x x f x x= = o

1


10/10

12% Eo the co+osite function( )( )f g xo

and ( )g x , find( )f x

G

e% ( ) ( )2( ) sin 1f g x x= +o

,2( ) 1g x x= +

f% ( )( )f g x x=o

,2( ) 1g x x= +

g% ( )( ) 4 , ( )f g x x g x x= =o

h%

( )

11

( )1

1 1

xf g x

x

+=

+

o

,

1( ) 1g x

x= +

2

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