p. 90
In this section we will show how combinations of familiar functions together with
some function operations can be used to construct more complex functions.
Arithmetic Combinations of Functions
If f and g are functions, then the functions f + g, f – g, f•g and f/g are defined by
(f + g)(x) = f(x) + g(x)
(f•g(x)) = f(x) • g(x)
(f – g)(x) = f(x) – g(x)
(f/g)(x) = f(x)/ g(x)
Note: Both f(x) and g(x) must be defined for any of these arithmetic operations
to be defined at x.
In other words, the domains of f + g, f – g, f•g, f/g must consist of those
real numbers that are common to both the domains of f and g.
Ex 1: Let f and g be defined by the table below. Show that f and g are functions,
and then find f + g, f – g, f•g, and f/g.
x
-3
-2
-1
0
1
2
3
f(x)
-1
1
2
-3
0
7
5
g(x)
0
-2
4
8
-1
3
5
x
f(x)
g(x)
-3
-2
-1
0
1
2
3
-1
1
2
-3
0
7
5
0
-2
4
8
-1
3
5
(f + g)(x)
-1
-1
6
5
-1
10
10
(f – g)(x)
-1
3
-2
-11
1
4
0
(f•g)(x)
0
-2
8
-24
0
21
25
(f/g)(x)
Not defined
-½
½
-3/8
0
7/3
1
1
Ex 2: Let f(x) = 2
and g ( x) 
x 1
x
x2
Find f + g, f – g, fg, f/g and give
their domains.
Solution:
Write the domains of f(x) and g(x).
D  (,1)  (1,1)  (1, )
f
D  (2, )
g
1
x

x 1
x2
1
x
( f  g )( x) 

x 1
x2
( f  g )( x) 
2
D
2
( f  g )( x) 
1
x

x 1 x  2
2
D
f  g , f  g , fg
 (2,1)  (1,0)  (0,1)  (1, )
f /g
1
1
x2
x2
( f / g )( x)  x  1 


x
x 1
x
x( x  1)
x2
2
2
 (2,1)  (1,1)  (1, )
2
Vertical Compression and Elongation of Graphs
1. For any constant c > 0, the basic shape of the graph of y = c•g(x) is the
same as the graph of y = g(x), but with a change in the vertical scale.
The domains of cg and g are the same and the graphs have the same
x-intercepts.
2. For c > 1, the graph of y = c•g(x) is a vertical elongation, and for
0 < c < 1 it is a vertical compression of the graph of y = g(x).
3. The graph of y = -g(x) is the reflection about the x-axis of the graph of
y = g(x).
For c < -1, the graph of y = c•g(x) is a vertical elongation and reflection
of the graph of y = g(x).
For -1 < c < 0, the graph of y = c•g(x) is a vertical compression and reflection
of the graph of y = g(x).
Look at p. 86-87 at graphs with different values of c.
Graphs of the Reciprocal Function
Let
f ( x) 
1
g ( x)
be the reciprocal of g(x).
1. f(x) is undefined when g(x) = 0.
2. f(x) and g(x) have the same value when g(x) = 1 or g(x) = -1
3. f(x) and g(x) have the same sign.
4. The magnitude of f(x) is large when the magnitude of g(x) is small.
5. The magnitude of f(x) is small when the magnitude of g(x) is large.
Ex 3: Use the graph of g(x) = x + 5 to determine the graph of
On board.
f ( x) 
1
x5