1.7_bzpc4e
TRANSCRIPT
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Section 1.7
Combinations of Functions:Composite Functions
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The Domain of a Function
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( ) ( )
( ) ( )
( ) ( ) ( )
2
2
3x+5g(x)=
x 4 5
x 4 5 0
5 1 0
5 0 1 0
x=5 x=-1
: - ,-1 1,5 5,
x
x
x x
x x
Domain
= + =
= + =
)
( ) 2 5
2 5 02x 5
5x
2
5: ,
2
h x x
x
Domain
=
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( )
2( ) 7
- ,
f x x x
Domain
=
Domains of Other Functions
( )
( ) 4 5
: ,
f x x
Domain
=
( )
3( )
: ,
f x x
Domain
=
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Example
Find the domain of the function4x-1
f(x)=3x+2
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Example
Find the domain of the functionf(x)= 4x-1
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The Algebra of Functions
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( ) ( )
( )( ) ( )
( )
2
2
fIf the function can beg
simplified, determine the domain
before simplifying.
Example;
f(x)= x and g(x)=x-2 2
f fx in ; Domain of : , ,2 22
g g
2 2f 22
g 22
x xxx x
x x
+ = = = +
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The domain of f+g is the set of all real numbers that
are common to the domain of f and the domain of g.
Thus we must find the domains of f and g before
finding their intersection.
Suppose ( ) x+3 af x = nd g(x)= x-2 then(f+g)(x)= x+3 2
Now for their domains.
( ) x+3 g(x)= x-2
3 0 x-2 0
3 x 2
So the domain for the sum of the functions
is x 2 which in inter
x
f x
x
x
+
=+
[ )val notation is 2,Continued on next slide
Determining Domains When Adding or
Subtracting Functions
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Continuation of the same problem.
4 3 2 1 1 2 3 4 5 6 7 8 9 1
2
1
1
2
3
4
5
6
x
y
[ )
The graph of (f+g)(x)= x+3 2
confirms that the domain of this functio n is 2,
x+
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The domain of f g is the set of all real numbers that
are common to the domain of f and the domain of g.
Thus we must find the domains of f and g before
finding their intersection.
5Suppose ( ) an
xf x
=
( )
3d g(x)= then
x-2
15
(fg)(x)= x x-2
Now for their domains.
5 3( ) g(x)=
x x-2
0 x-2 0x 2
So the domain for the product of the functions
is x 0, x 2 which in
f x
x
=
( ) ( ) ( )
interval notation is
- ,0 0, 2 2,
4 3 2 1 1 2 3 4
Determining Domains when Multiplying Functions
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Example2If f(x)=5x-1 g(x)=x 2 1 Find each of the following:
(f+g)(x)
(f-g)(x)
(fg)(x)
f( )
g
x
x
+
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Example
2
If f(x)=5x-1 g(x)=5x 9 2 Find the domainof the following:
(fg)(x)
f( )
g
x
x
+
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Example
1 1
If f(x)= g(x)= Find the domain2 1
of the following:
(fg)(x)
f( )
g
x x
x
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Example
If f(x)= x-1 g(x)= x-6 Find the domain for:
(f-g)(x)
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Composite Functions
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f(g(x))=0.85x - 300
We read this equation as "f of g of x is equal to 0.85x-300."
We call f(g(x)) the composition of the function f with g, or acomposite function. This composite function is written f go
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( ) ( ) ( )The domain of f g is ,0 0,3 3, o
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Example
( ) ( )
3 2
Given f(x)= and g(x)= .x-4 x
a. Find f g b. Find the domain of f gxo o
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Example
( ) ( )
2
Given f(x)= and g(x)= x.x-3
a. Find f g b. Find the domain of f gxo o
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Decomposing Functions
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Example
( )4
2
Express h(x) as a composition of two functions:
( ) 6 5h x x x= +
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Example
2
Express h(x) as a composition of two functions:1
( )9 64
h xx
=
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(a)
(b)
(c)
(d)
2
Find the domain of the function
3x-1f(x)=x 6 7x
( ) ( ) ( )
( ) ( ) ( )( ] [ )
( ] [ ] [ )
, 1 1,7 7,
,1 1,7 7,
, 1 7,
, 1 1,7 7,
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(a)
(b)
(c)
(d)
2
If f(x)=3x-1 and g(x)=x ,Find (f+g)(x)
2
2
2
2
3 1
(3 1)
3 1
3 1
x
x
x x
x x
+
+
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(a)
(b)
(c)
(d)
( ) ( )Find the domain of f g
if f(x)= x-4 and g(x)=4 3
x
x
o
[ )
[ )
( ]
( ]
0,
3,
, 3
,0