Sullivan Algebra and
Trigonometry: Section 3.5
Objectives
• Form the Sum, Difference, Product, and
Quotient of Two Functions
• Form the Composite Function and Find Its
Domain
If f and g are functions, their sum f + g is the
function given by
(f + g)(x) = f(x) + g(x).
The domain of f + g consists of the numbers x that
are in the domain of f and in the domain of g.
If f and g are functions, their difference f - g is
the function given by
(f - g)(x) = f(x) - g(x).
The domain of f - g consists of the numbers x that
are in the domain of f and in the domain of g.
Their product f  g is the function given by
(f  g)(x) = f(x)  g(x)
The domain of f  g consists of the numbers x that
are in the domain of f and in the domain of g.
Their quotient f / g is the function given by
(f / g)(x) = f(x) / g(x), g(x)  0
The domain of f / g consists of the numbers x for
which g(x) 0 that are in the domain of f and in
the domain of g.
Example: Define the functions f and g as follows:
f ( x)  x  3
g ( x)  x 2  16
Find each of the following and determine the
domain of the resulting function.
a.) (f + g)(x) = f(x) + g(x)

x  3  x  16
2
The domain of f consists of all real numbers x for
which x > 3; the domain of g consists of all real
numbers.
The domain of f + g is {x | x > 3}.
b.) (f + g)(x) = f(x) + g(x)


 x  3  x 2  16

x  3  x 2  16
The domain of f - g is {x | x > 3}.
c.) ( f  g )(x)= f(x)g(x)


  x  3 x 2  16
The domain of f  g is {x | x > 3}.
f ( x)  x  3
g ( x )  x  16
2
f ( x)
d.) ( f / g )( x ) 
g( x)
x3
 2
x  16
We must exclude x = - 4 and x = 4 from the
domain since g(x) = 0 when x = 4 or - 4.
The domain of f / g is {x | x > 3, x  4}.
Given two function f and g, the composite
function, denoted by f o g (read as “f
composed with g”) is defined by
f
g ( x)  f g x 
The domain of f o g is the set of all
numbers x in the domain of g such that
g(x) is in the domain of f.
Example: Given the functions f and g, find
(f o g)(2)
f
g 2  f g2
2
 f 6
 6 1

g ( x )  3x
g ( 2)  6
 35

f ( x)  x  1
2
Example: Given the functions f and g, find the
domain of f o g.
1
g ( x) 
x2
f ( x)  x
The domain of f o g consists of those x in the
domain of g, thus, x = - 2 is not in the domain of
the composite function.
Furthermore, the domain of f requires that g ( x)  0
1
So:
 0  x  2
x2
is {
The domain of f
g
x | x > -2}.
Example: Given the functions f and g, find f
1
g ( x) 
x2
f ( x)  x
f
g x   f gx 
1 

 f

 x  2

1
x2

1
x2
o g.