Section 3.6
Combining Functions
Let f and g be two 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)
f 
f ( x)
4. Quotient:  ( x) 
, g ( x)  0
g ( x)
g
The domains are the set of real numbers common to the domain of f and g (in f , g can’t be
g
equal to 0).
Example 1: Let f ( x)  x 2  3x  1 and g ( x)  3 x  10 . Find:
a. (g – f)(x)
b. (f + g)(x)
c. (f + g)(1)
d. (f g)(0)
Section 3.6 – Combining Functions
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e. (gg)(x)
f.
g (0)
f (0)
g. f(0) + g(-1)
h. g(3) + g(-3)
Section 3.6 – Combining Functions
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Example 2: Let f ( x) 
a. Find f ( x)  g ( x).
1
1
and g ( x) 
.
10 x
15 x
b. Find ݂ሺ‫ݔ‬ሻ݃ሺ‫ݔ‬ሻ
Example 3: Perform the indicated operation:
5
1
9


6 x 4 x 8x
Section 3.6 – Combining Functions
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The Composition of Functions
The composition of the function f with g is denoted by f  g and is defined by the equation
( f  g )( x)  f ( g ( x))
The domain of the composition f  g is the set of all x such that x is in the domain of g (the
“inside” function) and g(x) is in the domain of f (the “outside” function).
Example 4: Let f ( x) 
4
x2
and
g
(
x
)


 5 x . Find:
 x3  1
2
a. ( g  f )(0)
First:
Second:
b. f ( f (1))
First:
Second:
Section 3.6 – Combining Functions
4
2
Example 5: Let f ( x)  x  1 and g ( x )  2 x  5 , find:
a. ( f  g )( x ) .
b.
 g  f  x 
c. f  f  x  
d.
 f  f  0 
Section 3.6 – Combining Functions
5
Example 6: Let f ( x)  5  x and g ( x)  4  x 2 , find
a.  g  f  ( x) .
b. g ( f (3))
Section 3.6 – Combining Functions
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