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Hawkes Learning Systems:
College Algebra
Section 4.5: Combining Functions
HAWKES LEARNING SYSTEMS
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Objectives
o
o
o
o
Combining functions arithmetically.
Composing functions.
Decomposing functions.
Interlude: recursive graphics.
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Combining Functions Arithmetically
Addition, Subtraction, Multiplication and Division of Functions
1.  f  g  x   f  x   g  x 
2.  f  g  x   f  x   g  x 
3.  f  g  x   f  x   g  x 
f  x
 f 
x

, provided that g  x   0
4.    
g  x
g
The domain of each of these new functions consists of the common
elements (or the intersection of elements) of the domains of f and g
individually.
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Example 1: Combining Functions Arithmetically
Given that f  x   2 x 2  3x  4 and g  x   2 x solve:
a.  f  g  x 
Remember that  f  g  x   f  x   g  x  .
 f  x  g  x
 2 x 2  3x  4   2 x 
 2 x2  x  4
Continued on the next slide…
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Example 1: Combining Functions
Arithmetically (cont.)
Given that f  x   2 x 2  3x  4 and g  x   2 x solve:
b.  f  g  x 
 f  x  g  x
  2 x 2  3 x  4   2 x 
 4 x3  6 x 2  8 x
Remember that
 f  g  x  f  x   g  x  .
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Example 2: Combining Functions Arithmetically
Given that f  2   4 and g  2   3 find a. and b.
a.  f  g  2   f  2   g  2 
Remember that  f  g  x   f  x   g  x  .
  4    3
 7
f  2
 f 
b.    2  
g  2
g
4

3
f  x
 f
Remember that    x  
.
 g
g  x
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Example 3: Combining Functions Arithmetically
Based on the graphs of f and g below, determine the
f 
f
domain of and evaluate   1 .
g
g
We can observe that g is 0 when x = –2 and
x = 2. The domain of both f and g individually is
the entire set of real numbers, so the domain of
f
is  , 2   (2,2)   2,  . Also based on
g
the graphs it appears that f 1  1 and g (1)  3 ,
 f 
1
so   1  .
3
g
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Composing Functions
Composing Functions
Let f and g be two functions. The composition of f and
g, denoted f g , is the function defined by
 f g  x   f  g  x  .
The domain of f g consists of all x in the domain of g
for which g(x) is in turn in the domain of f. The function
f g is read “f composed with g,” or “f of g.”
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Composing Functions
Caution!
Note that the order of f and g is important. In general,
we can expect the function f g to be different from
the function g f . In formal terms, the composition of
two functions, unlike the sum and product of two
functions, is not commutative.
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Composing Functions
The diagram below is a schematic of the composition of
two functions. The ovals represent sets, with the
leftmost oval being the domain of the function g. The
arrows indicate the element that x is associated with by
the various functions. f g
g
x
f
g  x
f  g  x 
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Copyright © 2011 Hawkes Learning Systems.
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Example 4: Composing Functions
Given f  x   x3 and g  x   x  5, find:
First, we will find g(6) by
a.  f g  6 
g  6   6  5  11
f
g  6   f  g  6  
 f 11
 113
 1331
replacing x with 6 in g(x).
Next, we know that f composed
with g can also be written
f  g  6  . Since we already
evaluated g(6), we can insert the
answer to get f(11).
Continued on the next slide…
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Example 4: Composing Functions (cont.)
Given f  x   x3 and g  x   x  5, find:
Again, we know by definition
b.  f g  x   f  g  x  
that  f g  x   f  g  x  .
 f  x  5
Note: since we solved for the
  x  5
3
 x3  15x 2  75x  125
variable x we should be able to
plug 6 into x and get the same
answer as in part a. Verify this.
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Example 5: Composing Functions
Let f  x   x 2  6 and g  x   x . Simplify the compositions
and find the domains for:
a. f g
b. g f
 f  g  x 
 f
 x
 x6
Note: f g  g f .
 g  f  x 
 g  x2  6
 x2  6

Domain:  0, 
Domain: ,  6    6, 
Note: only non-negative numbers
can be plugged into g. Thus, the
domain is all positive real numbers.
The domain of g f must be any x
such that x2  6  0 since x 2  6 is
under a radical.

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Decomposing Functions
Often functions can be best understood by recognizing
them as a composition of two or more simpler
3
functions. For example, the function h  x    x  2  can
be thought of as the composition of two or more
functions.
Note: if f  x   x3 and g  x   x  2 then:
f
g  x   f  g  x    f  x  2 
  x  2
3
 h( x).
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Decomposing Functions
Ex: The function f  x   3 3x 2  5 can be written as a
composition of functions in many different ways. Some of
the decompositions of f(x) are shown below:
3
g
x

x


a.
g  h  x    g  3 x 2  5   3 3x 2  5  f  x 
h  x   3x 2  5
b. g  x   3 x  5
h  x   3x 2
c. g  x   3 x
h  x   3x  5
i  x  x
2
g  h  x    g  3 x 2   3 3x 2  5  f  x 

g h i  x 


 g h  x2 
 3 3x 2  5

 g  3x 2  5
 f  x
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Example 6: Decomposing Functions
Decompose the function f  x   x 4  5  1 into:
a. a composition of two functions
g  x  x 1
4
4
g
h
x

x
 5  1  f  x
    g  x  5
4
h x  x  5
b. a composition of three functions
g  x  x 1
h x  x  5
i  x   x4


  
g h  i  x    g h x4

 g x4  5

 x4  5  1  f  x
Note: These are NOT the only possible solutions for the
decompositions of f(x)!
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Interlude: Recursive Graphics
Recursion
Recursion refers to using the output of a function as its
input and repeating this process a certain number of
times. In other words, recursion is the composition of a
function with itself some number of times.
Some notations:
If f is a function, f 2  x  is used in this context to stand for
2
f  f  x   , or  f f  x  (not  f  x   !) Similarly, f 3  x 
stands for f f  f  x   , or  f f f  x  , and so on. The
functions f 2  x  , f 3  x  ,… are called iterates of f, with f n
being the n ͭ ͪ iterate of f.

