Chapter 1
Functions and Their Graphs
Warm Up 1.5
Given f (x) = 2x – 3 and g(x) = x2 – 1, find:
 f (x) + g(x) =
 f (x) – g(x) =
 f (x) · g(x) =
 f (x) / g(x) =
2
1.5 Combinations of Functions
Objectives:
Add, subtract, multiply, and divide functions.
Find compositions of one function with
another function.
Use combinations of functions to model and
solve real-life problems.
3
Vocabulary
Arithmetic Combination of Functions
Composition of Functions
4
Arithmetic Operations
Let f and g be two functions with overlapping
domains. Then, for all x common to both
domains:
1.
2.
3.
(f + g)(x) = f (x) + g(x)
(f – g)(x) = f (x) – g(x)
(f g)(x) = f (x) · g(x)
 f 
f x 
4.    x  
,
g x 
 g 
g x   0
5
Example 1
Let f (x) = 2x +1 and g (x) = x2 + 2x – 1.
Find (f – g)(x) algebraically and graphically.
Then find (f – g)(2).
6
Example 2
Find (f /g)(x) and (g/f )(x) for the functions given
by f  x  
x and g  x  
2
4x .
Specify the domain for each quotient.
7
Consider this …
 If you drop a pebble into a pond,
a circular ripple extends out from
the drop point. The radius of the
circle is a function of time. The
area enclosed by the circular
ripple is a function of the radius.
8
What If …?
Suppose that the radius is increasing at a constant
rate of 8 in/sec. Then
r = 8t
What is the radius at t = 5?
r = __________________
The formula for the area of the circular region is
A = ____________________
What is the area at t = 5?
A = ____________________
9
How Are the Variables Related?
Time is the input for the radius function.
Time
r(t)
Radius
A(r) or
A(r(t))
Area
And the radius is the input for the area function.
10
This means …
Area is a function of time through this chain:
Area depends on radius.
Radius depends on time.
Area is a composite function of time, A(r(t)).
11
Composition of Functions
The composition of the function f with the
function g is
f
 g  x   f  g  x 
12
Example 3
Find the following values
of the using the values of f
and g found in the table.
a. Find f (g(2).
b. Find g(f (2).
x
1
2
3
4
5
6
f (x)
3
4
6
2
0
1
g (x)
5
3
2
1
7
4
Example 4
Let f ( x ) 
x , x  0 and g ( x )  x  1, x  1 .
Find  f  g  x  .
Then find, if possible,  f  g  2  and
f
 g  0  .
14
Homework 1.5.1
Worksheet 1.5
#1, 7, 9, 17, 23, 35, 37, 51, 53, 73, 74
15
Warm Up 1.5
 A square concrete foundation
was prepared as a base for a large
cylindrical gasoline tank as
r
shown.
a. Write the radius r of the tank as
a function of the length x of the
sides of the square.
b. Write the area A of the circular
base of the tank as a function of
the radius r.
16
In-Class Assignment
Work with a partner to complete the worksheet
“Domain of Composite Functions”.
17
Domain of Composite Functions
The domain of f (g(x)) is the set of all x in the
domain of g such that g(x) is in the domain of f.
The outputs of g must be restricted so that they
are in the domain of f.
18
Finding the Domain
Let f ( x ) 
1
and g ( x )  x  1 .
x
Find the domain of  f  g  x  .
The outputs of g can be any real number.
But the domain of f is restricted to all real numbers ≠ 0.
Therefore, g  x   0 or x   1 .
So, domain of  f  g  is all real numbers except x   1 .
19
Example 4
Find the domain of the composition  f  g  x  for the
functions given by f  x   x  9 and g  x  
2
9x .
2
20
Example 5
Does  f  g  x    g  f
 x  ?
a. Let f  x   x  2 and g  x   4  x 2 .
b. Let f  x   2 x  3 and g  x  
1
2
x  3 .
21
Special Case
If (f ○g)(x) = (g ○ f)(x),
then
(f ○g)(x) = x
and
(g ○ f)(x) = x.
22
Application Problem
The number N of bacteria in a refrigerated food is given
by N(T) = 20T2 – 80T + 500, 2 ≤ T ≤ 14 where T is the
temperature of the food in ˚C.
When the food is removed from the refrigerator, the
temperature of the food is given by T(t) = 4t + 2, 0≤ t ≤ 3
where t is the time in hours.
a. Find the composition N(T(t)) & interpret its meaning.
b. Find the number of bacteria in the food when t = 2
hours.
c. Find the time when the bacterial count reaches 200.
23
Homework 1.5.2
Worksheet 1.5
# 39 – 49 odd, 63 – 71 odd, 77
24