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Functions
Copyright © Cengage Learning. All rights reserved.
2.6
Combining Functions
Copyright © Cengage Learning. All rights reserved.
Objectives
► Sums, Differences, Products, and Quotients
► Composition of Functions
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Sums, Differences, Products,
and Quotients
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Sums, Differences, Products, and Quotients
Two functions f and g can be combined to form new
functions f + g, f – g, fg, and f/g in a manner similar to the
way we add, subtract, multiply, and divide real numbers.
For example, we define the function f + g by
(f + g)(x) = f(x) + g(x)
The new function f + g is called the sum of the functions
f and g; its value at x is f(x) + g(x).
Of course, the sum on the right-hand side makes sense
only if both f(x) and g(x) are defined, that is, if x belongs to
the domain of f and also to the domain of g.
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Sums, Differences, Products, and Quotients
So if the domain of f is A and the domain of g is B, then the
domain of f + g is the intersection of these domains, that is,
A  B. Similarly, we can define the difference f – g, the
product fg, and the quotient f/g of the functions f and g.
Their domains are A  B, but in the case of the quotient we
must remember not to divide by 0.
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Example 1 – Combinations of Functions and Their Domains
Let f(x) =
and g(x) =
(a) Find the functions f + g, f – g, fg, and f/g and their
domains.
(b) Find (f + g)(4), (f – g)(4), (fg)(4), and (f/g)(4).
Solution:
(a) The domain of f is {x|x  2}, and the domain of g is
{x|x  0}. The intersection of the domains of f and g is
{x|x  0 and x  2} = [0, 2)  (2,
)
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Example 1 – Solution
cont’d
Thus, we have
Domain {x | x  0 and x  2}
Domain {x | x  0 and x  2}
Domain {x | x  0 and x  2}
Domain {x | x  0 and x  2}
Note that in the domain of f/g we exclude 0 because
g(0) = 0.
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Example 1 – Solution
cont’d
(b) Each of these values exist because x = 4 is in the
domain of each function.
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Sums, Differences, Products, and Quotients
The graph of the function f + g can be obtained from the
graphs of f and g by graphical addition. This means that
we add corresponding y-coordinates, as illustrated in the
next example.
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Example 2 – Using Graphical Addition
The graphs of f and g are shown in Figure 1. Use graphical
addition to graph the function f + g.
Figure 1
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Example 2 – Solution
We obtain the graph of f + g by “graphically adding” the
value of f(x) to g(x) as shown in Figure 2.
Graphical addition
Figure 2
This is implemented by copying the line segment PQ
on top of PR to obtain the point S on the graph of f + g.
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Composition of Functions
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Composition of Functions
Now let’s consider a very important way of combining two
functions to get a new function. Suppose f(x) =
and
g(x) = x2 + 1. We may define a new function h as
h(x) = f(g(x)) = f(x2 + 1) =
The function h is made up of the functions f and g in an
interesting way: Given a number x, we first apply the
function g to it, then apply f to the result.
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Composition of Functions
In this case, f is the rule “take the square root,” g is the rule
“square, then add 1,” and h is the rule “square, then add 1,
then take the square root.”
In other words, we get the rule h by applying the rule g and
then the rule f. Figure 3 shows a machine diagram for h.
The h machine is composed of the g machine (first)
and then the f machine.
Figure 3
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Composition of Functions
In general, given any two functions f and g, we start with a
number x in the domain of g and find its image g(x). If this
number g(x) is in the domain of f, we can then calculate the
value of f(g(x)).
The result is a new function h(x) = f(g(x)) that is obtained
by substituting g into f. It is called the composition (or
composite) of f and g and is denoted by f  g (“f composed
with g”).
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Composition of Functions
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. In other words, (f  g)(x)
is defined whenever both g(x) and f(g(x)) are defined.
We can picture f  g using an arrow diagram (Figure 4).
Arrow diagram for f  g
Figure 4
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Example 3 – Finding the Composition of Functions
Let f(x) = x2 and g(x) = x – 3.
(a) Find the functions f  g and g  f and their domains.
(b) Find (f  g)(5) and (g  f)(7).
Solution:
(a) We have
(f  g)(x) = f(g(x))
= f(x – 3)
Definition of f  g
Definition of g
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Example 3 – Solution
= (x – 3)2
and (g  f)(x) = g(f(x))
cont’d
Definition of f
Definition of g  f
= g(x2)
Definition of f
= x2 – 3
Definition of g
The domains of both f  g and g  f are
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Example 3 – Solution
cont’d
(b) We have
(f  g)(5) = f(g(5)) = f(2) = 22 = 4
(g  f)(7) = g(f(7)) = g(49) = 49 – 3 = 46
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Composition of Functions
You can see from Example 3 that, in general, f  g  g  f.
Remember that the notation f  g means that the function g
is applied first and then f is applied second.
It is possible to take the composition of three or more
functions. For instance, the composite function f  g  h is
found by first applying h, then g, and then f as follows:
(f  g  h)(x) = f(g(h(x)))
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Example 5 – A Composition of Three Functions
Find f  g  h if f(x) = x/(x + 1), g(x) = x10 , h(x) = x + 3.
Solution:
(f  g  h)(x) = f(g(h(x)))
Definition of f  g  h
= f(g(x + 3))
Definition of h
= f((x + 3)10)
Definition of g
Definition of f
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Note that the domain of f o g is the set of all real
numbers x such that x is in the domain of the function
g and g(x) is in the domain of the function f.
f ( x)  1  x 2
2
g ( x) 
x
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Composition of Functions
So far, we have used composition to build complicated
functions from simpler ones.
But in calculus it is useful to be able to “decompose” a
complicated function into simpler ones, as shown in the
following example.
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Example 6 – Recognizing a Composition of Functions
Given F(x) =
F = f  g.
find functions f and g such that
Solution:
Since the formula for F says to first add 9 and then take the
fourth root, we let
g(x) = x + 9 and f(x) =
Then
(f  g)(x) = f(g(x))
Definition of f  g
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Example 6 – Solution
= f(x + 9)
cont’d
Definition of g
Definition of f
= F(x)
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