Combinations of Functions Composite Functions. Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with overlapping domains.

Combinations of Functions

Composite Functions

Sum, Difference, Product, and Quotient of Functions

• Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as:

1. Sum2. Difference3. Product

4. Quotient

)()())(( xgxfxgf

)()())(( xgxfxgf

)()())(( xgxfxfg

0)(,)(

)()(

xg

xg

xfx

g

f

Example 1

• Given and

• Find the sum and difference of the functions.

12)( 2 xxxg12)( xxf

Example 1

• Given and

• Find the sum and difference of the functions.

12)( 2 xxxg12)( xxf

xxxxxxgxfxgf 4)12()12()()())(( 22

Example 1

• Given and

• Find the sum and difference of the functions.

12)( 2 xxxg12)( xxf

xxxxxxgxfxgf 4)12()12()()())(( 22

2)12()12()()())(( 22 xxxxxgxfxgf

Example 1

• Given and

• Find the product and quotient of the functions.

12)( 2 xxxg12)( xxf

Example 1

• Given and

• Find the product of the functions.

12)( 2 xxxg12)( xxf

15212242

)12)(12()()())((23223

2

xxxxxxx

xxxxgxfxfg

Example 3

• Given and

• Find the quotient of the functions.

• The domain of g(x) is all numbers except 3, -1. Even though we canceled, this is still the domain.

32)( 2 xxxg1)( xxf

3

1

)3)(1(

1

32

1

)(

)(2

xxx

x

xx

x

xg

xf

Example 3

• Given and

• Find the quotient of the functions.

• The domain of f(x) is all numbers except -1. Even though we canceled, this is still the domain.

32)( 2 xxxg1)( xxf

31

)3)(1(

1

32

)(

)( 2

xx

xx

x

xx

xf

xg

You Try

• Given and

• Find the sum and difference of the functions.

23)( 2 xxxg23)( xxf

You Try

• Given and

• Find the sum and difference of the functions.

23)( 2 xxxg23)( xxf

443)23()23()()())(( 22 xxxxxxgxfxgf

xxxxxxgxfxgf 23)23()23()()())(( 22

You Try

• Given and

• Find the product and quotient of the functions.

23)( 2 xxxg23)( xxf

You Try

• Given and

• Find the product and quotient of the functions.

• The domain is still

23)( 2 xxxg23)( xxf

4839)23)(23()()())(( 232 xxxxxxxgxfxgf

11

)1)(23(23

2323

2)(/)())((

xxx

xxx

xxgxfxgf

1,32 x

Composition of Functions

• Definition of The Composition of Two Functions.

• The composition of the function f with the function g is

• The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.

)( gf

))(())(( xgfxgf

Example 4

• Given and , find the following.

a)

This means go to the f function and replace x with the g function.

24)( xxg 2)( xxf

))(( xgf

Example 4

• Given and , find the following.

a)

This means go to the f function and replace x with the g function.

24)( xxg 2)( xxf

))(( xgf

22 62)4())(( xxxgf

Example 4

• Given and , find the following.

a)

This means go to the g function and replace x with the f function.

24)( xxg 2)( xxf

))(( xfg

xxxxxxfg 4)44(4)2(4))(( 222

Example 4

• Given and , find the following.

b)

This means go to the f function and replace x with -2. Take that answer and put it into g.

24)( xxg 2)( xxf

)2)(( fg

022)2( f

404)0( 2 g

Example 4

• Given and , find the following.

b)

You could also go to your answer for and replace x with a -2.

24)( xxg 2)( xxf

)2)(( fg

))(( xfg

4)2(4)2(4))(( 22 xxxfg

You Try

• Given and , find the following.

a)

b)

c)

d)

xxxg 2)( 2 2)( xxf

))(( xgf

))(( xfg

)2)(( gf

)3)(( fg

You Try

• Given and , find the following.

a)

b)

c)

d)

xxxg 2)( 2 2)( xxf

))(( xgf

))(( xfg

222)2( 22 xxxx

xxxx 2)2(2)2( 22

)2)(( gf

)3)(( fg 3)3(2)3( 2

62)2(2)2( 2

Finding the Domain of the Composition of Functions

• Given and , find the composition f(g(x)). Then find the domain of f(g(x)).

9)( 2 xxf 29)( xxg

Finding the Domain of the Composition of Functions

• Given and , find the composition f(g(x)). Then find the domain of f(g(x)).

• We take the f function and replace x with the g function.

9)( 2 xxf 29)( xxg

99))((2

2 xxgf

22 99 xx

Example 5

• So the composition , and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition.

, so its domain is where

2))(( xxgf

29)( xxg 09 2 x

Example 5

• So the composition , and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition.

, so its domain is where• We need to do sign analysis. ___-___|___+___|___-___ -3 3

2))(( xxgf

29)( xxg 09 2 x

]3,3[

Class work

• Page 238

• 6, 8, 14, 16, 20, 22, 36, 42

Homework

• Page 238

• 5-19 odd

• 35,37,43