Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 2.6...

Chapter 2Functions and

Graphs

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2.6 Combinations ofFunctions; Composite Functions


• Find the domain of a function.• Combine functions using the algebra of functions,

specifying domains.• Form composite functions.• Determine domains for composite functions.• Write functions as compositions.

Objectives:


Finding a Function’s Domain

If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in a square root of a negative number.


Example: Finding the Domain of a Function

Find the domain of the function

Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero.

We exclude 7 and – 7 from the domain of g.

The domain of g is

2

5( )

49x

g xx

2 49 0x 2 49x

49x

7x ( , 7) ( 7,7) (7, )


The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions. The sum f + g, the difference, f – g, the product fg, and the quotient

are functions whose domains are the set of all real numbers common to the domains of f and

defined as follows:

1. Sum:

2. Difference:

3. Product:

4. Quotient:

fg

( ),f gg D D

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

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

( )( )

f f xx

g g x


Example: Combining Functions

Let and Find each of the following:

a.

b. The domain of

The domain of f(x) has no restrictions.

The domain of g(x) has no restrictions.

The domain of is

( ) 5f x x 2( ) 1.g x x

( )( )f g x 2( 5) ( 1)x x 2 6x x

( )( )f g x

( )( )f g x ( , )


The Composition of Functions

The composition of the function f with g is denoted

and is defined by the equation

The domain of the composite function is the set of all x such that

1. x is in the domain of g and

2. g(x) is in the domain of f.

f g

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

f g


Example: Forming Composite Functions

Given and find

f g( ) 5 6f x x 2( ) 2 1,g x x x

( ( ))f g f g x 2(2 1)f x x 25(2 1) 6x x

210 5 5 6x x 210 5 1x x


Excluding Values from the Domain of

The following values must be excluded from the

input x:

If x is not in the domain of g, it must not be in the domain of

Any x for which g(x) is not in the domain of f must not be in the domain of

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

.f g

.f g


Example: Forming a Composite Function and Finding Its Domain

Given and

Find

4( )

2f x

x

1

( )g xx

( )( )f g x

( )( ) ( ( ))f g x f g x1

fx

41

2x

41

2

xx

x

41 2xx

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


Example: Forming a Composite Function and Finding Its Domain

Given and

Find the domain of

For g(x),

For

The domain of is

4( )

2f x

x

1

( )g xx

( )( )f g x

4( )( ) ,

1 2x

f g xx

0x

12

x

( )( )f g x 1 1, ,0 0,

2 2


Example: Writing a Function as a Composition

Express h(x) as a composition of two functions:

If and then

2( ) 5h x x

( )f x x 2( ) 5,g x x ( ) ( )( )h x f g x

Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 2.6...

Documents

Transcript of Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 2.6...