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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
