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Advanced Algebra
Name _____________________________________
2.6 Notes: Combinations of Functions: Composite Functions
Objective: In this lesson, you learned how to find arithmetic combinations and composition of functions.
Arithmetic Combinations of Functions
Just as two real numbers can be combined with arithmetic operations, two functions can be combined by the
operations of _______________ , _______________ , _______________, and _______________ to create
new functions.
Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions with overlapping domains. Then, for all x _______________________
3.
 g  x   __________________________________
Difference:  f  g  x   ____________________________
Product:  fg  x   _________________________________
4.
Quotient:
1.
2.
Sum:
f
 f 
   x   _________________________________
g
The domain of an arithmetic combination of functions f and g consists of all real number that are
________________________________________________________ . In the case of the quotient
f  x
g  x
,
there is the further restriction that _______________.
Example
Given
Example
Given
Find each of the arithmetic combinations below
and state their domains.
 f  g  x   ______________________
Find each of the arithmetic combinations below
and state their domains.
 f  g  x   ______________________
 f 
   x   _________________________
g
 f 
   x   _________________________
g
f  x   3x  2 and g  x   x 2  2 x  3
 f  g  x   ______________________
 fg  x   _________________________
f  x   x 2  1 and g  x   4  x
 f  g  x   ______________________
 fg  x   _________________________
Example
2
Evaluate the indicated function for f  x   3 x  4 and g  x   5  x .
f
 g  3  ______________________
f
 g  n  2   _____________________
 f 
   4   _________________________
g
 fg  2t   _________________________
Composition of Functions
The composition of the function f with the function g is
f
f
g  x   f  g  x   . The domain of
g  is the set ____________________ in the domain of ______ such that _________ is in the domain of
_____.
Example
2
Given f  x   2 x  1 and g  x   x  3 , find the following:
f
g  x 
g
Given f  x   x  25 and g  x  
2
f
g  x 
Given f  x  
f
g  x 
g  x 
f
g  1
25  x 2 , find the following and state the domain of each:
g
f  x 
x  9 and g  x   x 2 , find the following and state the domain of each:
 g f  x 
Given f  x   x  4 and g  x  
f
f  x 
1
, find the following and state the domain of each:
x
 g f  x 