Section 1.5
Combinations
of
Functions
Combinations of Functions
Arithmetic Combinations of Functions
The domain of an arithmetic combination of functions f and g consists
of all real numbers that are common to the domains of f and g. In the
Case of f(x)/g(x), there is the further restriction that g(x)  0.
Sum, Difference, Product and Quotient Functions
1. Sum
 f  g  x   f  x   g  x 
2. Difference
 f  g  x   f  x  g  x 
3. Product
 fg  x   f  x  g  x 
4. Quotient
 f / g  x   f  x  / g  x  ,
g  x  0
Combinations of Functions
Try these:
Given
f ( x)  x 2 , g ( x)  x  2
Find:
1. (f + g)(x)
 f  g  x   f  x   g  x 
 x2  x  2
3. (f g)(x)
 fg  x   f  x   g  x 
 x2  x  2
 x3  2 x 2
2. (f – g)(x)
 f  g  x   f  x  g  x 
 x2   x  2
 x2  x  2
4. (f / g)(x)
f  x
f 
x

, g  x  0
  
g  x
g
x2

, x2
x2
Combinations of Funcitons
Try these:
2
Given: f ( x)  x  1, g ( x)  x  5
Find:
1. (f + g)(3) = 8
2. (f – g)(-1) = - 8
3. (f g)(-2) = - 35
4. (g / f)(0) = -1/5
Combinations of Functions
Definition of Composition of Two Functions
The composition of the function f with the function g is
( 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.
Combinations of Functions
How to from the composite of two functions:
Given: If f  x   x  2 and g  x   4  x2 , find  f g  x.
Step 1: Think of f(x) as being the primary function and g(x) being the
secondary function. To form the composite substitute the
secondary function into the primary function.
f
g  x   f  g  x  
  4  x2   2
Step 2: Simplify the result.
f
g  x    x2  6
Definition of composite
Substitution of 4 – x2 for x in x + 2
Combinations of Functions
How to from the composite of two functions (cont.):
Given: If f  x   x  2 and g  x   4  x2 , find  g f  x.
Step 1: Think of g(x) as being the primary function and f(x) being the
secondary function. To form the composite substitute the
secondary function into the primary function.
g
f  x  g  f  x  
 4   x  2
2
Step 2: Simplify the result.
g
f  x  4   x2  4x  4
 4  x2  4 x  4
g
f  x   x2  4x
Definition of composite
Substitution of x + 2 for x in 4 – x2
Combinations of Functions
How to find the numeric value for the combination of two functions
Given: If f  x   3x  5 and g ( x)  5  x, find  f g  2  .
Step 1: Find g(2).
Step 2: Take the value for g(2) and
substitute it into f(x).
Step 3: The answer found in step 2 is the
numeric value for  f g  2
g  2  5  2  3
f  x   3  3  5  4
f
g  2  4
Combinations of Functions
How to find numeric values of composite graphically.
Given:
If f  x   x2  3and g ( x)  2 x  1, find  f g  0 .
Step 1: Graph both functions on individual graphs.
Step 2: Find g(0)
g(0) = 1
Step 3: Take the value
from Step 2 and
substitute It in for
x in f(x). Find its
value on the graph.
f(x) = x2 - 3
f(1) = -2
g(x) = 2x + 1
f
g  0  2
Combinations of Functions
Try these
Find  f g  x  and  g f  ( x) for the following :
1.
f  x   5x  4
g  x  4  x
2.
f  x  x  6
g  x   x 2 5
f
g
g  x   5x  24
f  x   5x
2
f
g
x

x
1

 
g
f  x   x  1, x  6
Combinations of Functions
What you should have learned:
1. To add, subtract, multiply, and divide functions
2. To find the composition of one function with
another function.