Transcript 6.1 Composite Functions In this section, we will study the following topics:  Evaluating composite functions  Finding domain and range of composite functions.

6.1 Composite Functions
In this section, we will study the following topics:

Evaluating composite functions

Finding domain and range of composite
functions
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Compositions of Functions
Definition of Composition of Two Functions
f

g  (x ) = f g  x 

The domain of  f g  is the set of all x in the domain of g such that g(x)
is in the domain of f.
The composition of functions  f g   x  means that you will
first evaluate x in the function g. Then you will take that
result and evaluate it in function f.
So you evaluate in one function and then the result in the
other.
Just be careful; ORDER MATTERS!
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Compositions of Functions
f
g  ( x)
g
f  ( x)
means evaluate x first in g, then the result in f.
means evaluate x first in f, then the result in g.
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Compositions of Functions
So if f ( x )  x 3  5 and g ( x )  x  2
then to find

f
Use the function
CLOSEST to the input
value FIRST
g    1
you would first evaluate g at x = –1:
g (  1)
 1  2  3
Then you would evaluate f using this result:
f (  3 )  (  3 )  5   27  5   3 2
3
Write out complete answer:

f
g  (  1)   3 2
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Compositions of Functions (continued)
Take a look*:
f
g  x  and g
x 
 g f  x 
G iv e n f ( x )  x - 3 a n d g ( x )  2 x  1, e v a lu a te
2
f
  x .
Solution:
f
g x 
 f g  x 
 f  2 x  1
g
f
g
  2 x  1  3



2


State the domain of each composite function.
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Example
G iv e n f ( x ) 
a)  f
x a n d g ( x )  3 x -1 , e v a lu a te :
2
g  x ,
b)
g
f
  x ,
and c ) g
 1
f   .
4
Solution:
a)
f
g x  
b)
g
f
x  
c)
g
1
f   
4
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Decomposition of Functions
Sometimes we decompose functions, which, I assure you,
has nothing to do with rotting flesh.
To decompose a function, we basically write a given
function as a composition of two or more functions.
Example
Express h ( x ) 
such that
2x  9
h(x) 

f
as a composition of two functions f and g
g . x 
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Example
Express
h( x) 
1
 x  2
3
as a composition of two functions
f and g such that h ( x )   f
g   x .
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End of Section 6.1
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