2.6
COMBINATION OF FUNCTIONS; COMPOSITE FUNCTIONS
If f x  2x  3 and g x   x 2  1 , then we can combine these functions to form the sum,
difference, product and quotient of them.
f x   g x  
f x   g x  
f x g x  
f x 

g x 
This one will work as long as g x  0 .
Example 1
Let f x  2 x  1 and g x   x 2  2 x  1 .
a)
Find  f  g x 
b)
Evaluate  f  g 3
b)
Evaluate  f  g 2
b)
Evaluate  fg 4
Example 2
Let f x  2 x  1 and g x   x 2  2 x  1 .
a)
Find  f  g x
Example 3
Let f x   x 2 and g x   x  3 .
a)
Find  fg x
Example 4
Let f  x   x and g x   4  x 2 .
a)
f
Find   x 
g
b)
g
Find   x 
f
c)
f
Find the domain of   x 
g
d)
g
Find the domain of   x 
f
Example 5
Use the graphs of f and g to draw the graph of hx   f  g x .
a)
b)
Composition of Functions
Another way of combining functions is to form the composition of one with the other.
If f x   x 2 and g x   x  1 , then the composition of f with g is as follows:
f g x   f x  1  x  1 .
2
This composition is denoted as f  g or
 f  g x .
It is read as “f composed with g.” It can
also be read as “f circle g.”
The composition of the function f with the function g is  f  g x . It means f g x .
Example 6
If f x   x  2 and g x   4  x 2 , find the following:
a)
 f  g x
b)
g  f x
c)
g  f  2
Example 7
Use the graphs of f and g to evaluate each function.
8
y = f(x)
6
4
2
-10
-5
5
-2
-4
y = g(x)
-6
-8
-10
Guided Practice
Your Turn
a)
f
 g  3
f
 g  4
b)
f
 g 2
f
 g 2
c)
f 
  6 
g
f 
  5
g
d)
 fg 4
 fg  2
e)
 f  g 2
 f  g 1
f)
g  f 0
g  f  1
10
15