LESSON 11 (1.5) COMBINATIONS OF FUNCTIONS
You should learn to:
1. Add, subtract, multiply and divide functions.
2. Find compositions of one function with another function.
3. Use combinations of functions to model and solve real-life problems.
Terms to know: sum, difference, product, and quotient (arithmetic combinations of functions), composition of
functions
1. Sum:
 f  g  x   f  x   g  x 
3. Product:
2. Difference:
 fg  x   f  x   g  x 
4.
Quotient:
 f  g  x   f  x   g  x 
f  x
 f 
, g  x  0
  x 
g  x
g
Example 1: Given f  x   x2  4 and g  x   x  3 , find:
a.
 g  f  x 
 g  f  x 
b.
 fg  x 
 fg  x 
g ( x)  f ( x)
f ( x)  g ( x)
( x  3)  ( x 2  4)
( x 2  4)( x  3)
x  3  x2  4
x3  3x 2  4 x  12
c.
g
   x
f 
g
  x
f 
g ( x)
f ( x)
 x2  x  1
x3
x2  4
Domain : x  2
Which of the functions in parts a - c have domain restrictions? List the domain restrictions.
C : The domain restictions are x  2
Example 2: Using the functions above, find each of the following.
a.
( f  g )(3)
b.
( fg )( 2)
( f  g )(3)
f (3)  g (3)
( fg )(2)
f (2)  g (2)
( x 2  4)  ( x  3)
( x 2  4)( x  3)
(32  4)  (3  3)
(9  4)  (0)
50
((2)  4)(2  3)
(4  4)(5)
(0)(5)
5
0
2
f
(4)
g
c. 
 f 
  (4)
g
f (4)
g (4)
x2  4
x3
42  4
43
16  4
1
12
When combining functions using addition, subtraction, or multiplication, whatever domain restrictions occur
in the components (parts) will occur in the combined functions.
Example 3: Given f  x  
x ( Domain : x  0 ) and g  x  
 gf  x  , and list the domain restrictions for each.
a.
 f  g  x  
b.
 gf  x  
1
x
D:
x0
1
x
 x
x
x
D:
x0
x
1
, ( Domain : x  0 ) find  f  g  x  and
x
When dividing functions, all domain restrictions occurring in the original functions will occur in the combined function.
However, new domain restrictions may occur (mainly because the denominator of the new function cannot equal zero).
Example 4: Given f  x   2  x and g  x   x , find:
g
x
 ( x) =
2 x
 f 
Domain : 2  x  0
 x  2
a. 
g
 1 =
f 
b. 
Domain : x  2
f 
  x =
g
e. 
2 x
x
Domain : x  2, x  0
g
1
c.    0  =
 f 
2 1
1
1
0
2
g
  3 =
 f 
d. 
0
1
NRS
 f 
  0 = 2
g
0
f. 
undefined
A composition of functions occurs when one function is “put into” another function. The composition
of function f with function g is symbolized by ( f g )( x ) or f ( g ( x)) and means that g ( x ) is
“put into” f ( x ) .
If f ( x)  2 x  3 and g ( x)  x 2 :
g  f  x  =
f ( g ( x)) 
f ( g ( x))  2 x  3
2
2 x2  3
3
23
3
1
g  f  x    (2 x  3)2
(2 x  3)(2 x  3)
4 x 2 6 x  6 x  9
4 x 2  12 x  9
Example 5: Given f ( x) 
a. ( f
x and g ( x)  x  1 , find the following. List the domain for each function.
g )( x) = f ( g ( x))
b. ( g
f )( x)  g ( f ( x))
f ( g ( x))  x  1
g ( f ( x))  x  1
f ( g ( x))  x  1
Domain : x  1  0
g ( f ( x))  x  1
Domain : x  0
Domain : x  1
To “decompose” a function, look for “inner” and “outer” functions. There is oftentimes more than one way to
“decompose” a function.


Example 6: Write each function as the composition of two functions. That is, h  x   f g  x  , where
a.
h  x 
1
x 2
b. h( x) 
2
g  x  x2  2
f  x 
1
x
or
or
2x  3
g  x   2x  3
x2
1
x2
f  x 
x
or
or
2x
x 3
Example 7: The weekly cost of producing x units in a manufacturing process is given by the function
C  x   48x  1150 . The number of units produced in t hours is given by x  t   40t .
a. Find and interpret  C x  (t ) .
C ( x(t ))  48(40(t ))  1150
C (t )  1920t  1150
The hourly production cost.
b. Find  C x  20 .
1920(20)  1150
$39,550
ASSIGNMENT 11 Pages 58-61 (7, 9, 13-19 odd, 35, 37, 39, 41, 43, 49, 51 (You don’t need to use a calculator to
verify or graph), 63, 65, 69, 83, 98 (put into point-slope form first, and then into slope-intercept form)); Page 25: (28);
Page 29: (93); Page 48: (7) Page A61: (101, 105, 151)