fall 2014 comp 2300 discrete structures for computation donghyun (david) kim department of...

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Fall 2014COMP 2300 Discrete Structures for Computation

Donghyun (David) KimDepartment of Mathematics and PhysicsNorth Carolina Central University

Chapter 7.3Composition of Functions

2Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Composition of Two Functions• Let and be

functions with the property that the range of f is a subset of the domain of g and Define a new function as follows:

where is read g circle f and is read g of f of x. The function is called the composition of f and g.

YXf : ZYg :

ZXfg :,))(())(( Xxxfgxfg all for

fg ))(( xfgfg

.YY


Composition of Functions De-fined by Formulas• Suppose two functions and

defined as below:

a. Find the compositions and .

b. Is ? Explain.gffg

:g:f

nnng

nnnf

all for

all for 12)(

)(

fg gf



defined as below:



:g:f

nnng

nnnf

all for

all for 12)(

)(

fg gf

nnngnfgnfg all for 11 2)()())(())(( nnnfngfngf all for 122)())(())((



defined as below:



:g:f

nnng

nnnf

all for

all for 12)(

)(

fg gf

nnngnfgnfg all for 11 2)()())(())(( nnnfngfngf all for 122)())(())((

2141 ))(())(( gffg


Composition of Functions De-fined on Finite Sets• Let

and . Define two functions and by the arrow diagram be-low.

• Draw the arrow diagram for . What is the range of ?

},,,,{},,,,{},,,{ edcbaYdcbaYX 321},,{ zyxZ YXf :

ZYg :

fg fg

1

2

3

abc

d

XY

Z

e

x

y

z

f g

Y’


Composition of Functions De-fined on Finite Sets – cont’• Let

and . Define two functions and by the arrow diagram be-low.

• Draw the arrow diagram for . What is the range of ? Range is {y, z}

Fall 2010 COMP 4605/5605 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

},,,,{},,,,{},,,{ edcbaYdcbaYX 321},,{ zyxZ YXf :

ZYg :

fg fg

abc

d

1

2

3

XY

e

Z

x

y

z

f g

Y’

1

2

3

X Z

x

y

z

fg


Composition with the Identity Function• If f is a function from a set X to a set Y, and

is the identity function on X, and is the identity function on Y, then

XI

YI

.)()( ffIbfIfa YX and

)())(()(

)())(()(

xfxfIxfI

xfxIfxIf

XX

XX

Roughly, this is because

More formal proof is given in the textbook.


Composing a Function with Its Inverse• If is a one-to-one and onto

function with inverse function then

,: XYf 1YXf :

YX IffbIffa 11 and )()(

)()())(()(,)(

)()())(()(,)(

yIxfyffyxfxyf

xIyfxffxyfyxf

Y

X

11

111

then if

then if

Roughly, this is because

More formal proof is given in the textbook.


Composition of One-to-One Functions• Suppose we have two one-to-one functions

f and g. Is their composite function one-to-one?


Composition of One-to-One Functions• Suppose we have two one-to-one functions f and

g. Is their composite function one-to-one? Yes!• Proof• Suppose we have we two different elements

such that

Since

we have Also, since f is one-to-one

has to be true. Since g is also one-to one,

has to be true. (Contradiction!)

21 and xx). ()( 21 xgfxgf

, and 2211 ))f(g(x)g(xf))f(g(x)g(xf

)g(x)g(x 21

).(( 21 ) xgfxgf

21 xx


Composition of Onto Functions• Suppose we have two onto functions

and . Is their composite function still onto?

YXf :ZYg :



and . Is their composite function still onto? Yes

. that such find

to possible is it , any given onto is :

zf)(x)(gXx

ΖzZXfg

YXf :ZYg :



and . Is their composite function still onto? Yes

• Proof• Since g is onto, there has to be such that

. Also, since f is onto, there has to be some such that . Therefore, it is true.

. that such find

to possible is it , any given onto is :

zf)(x)(gXx

ΖzZXfg

YXf :ZYg :

zyg )(Yy

Xx yxf )(

fall 2014 comp 2300 discrete structures for computation donghyun (david) kim department of...

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