Operations on Functions Section 1-8

1
Operations on FunctionsSection 1-8
2
Objectives

• I can add, subtract, multiply, and divide
  functions
• I can find the domains of newly formed functions

3
Chapter 1 Section 8 Arithmetic of Functions

• NOTATION

f(x) g(x) means
Add the f(x) function to the g(x) function.
4
Chapter 1 Section 8 Arithmetic of Functions

• NOTATION

f(x) - g(x) means
Subtract the g(x) function from the f(x) function.
5
Chapter 1 Section 8 Arithmetic of Functions

• NOTATION

(fg)(x) means
f(x) g(x)
Multiply f(x) times g(x)
6
Chapter 1 Section 8 Arithmetic of Functions

• NOTATION

(f/g)(x) means
f(x) g(x)
Divide f(x) by g(x)
7
Let f(x) 3x 4 g(x) 2x 3

• find
• f(x) g(x)
• f(x) g(x)
• f(x)g(x)
• d) f(x)/g(x)

5x 1
x 7
6x2 x - 12
8
Operations on Functions Addition (f g)(x)
f(x) g(x) Example If f(x) 3x 6 and g(x)
-3x2 3x 4 then (f g)(x) 3x 6 -3x2
3x 4 -3x2 6x 2 Subtraction (f g)(x)
f(x) g(x) Example Use the same functions as
above. Then (f - g)(x) 3x 6 (-3x2 3x
4 ) (f g)(x) 3x2 - 10
9
Operations on Functions Multiplication (fg)(x)
f(x)g(x) Example If f(x) 3x 6 and g(x)
-3x2 3x 4 then (fg)(x) (3x 6)(-3x2 3x
4) -9x3 9x2 12x 18x2 18x 24 -9x3
27x2 6x - 24 Division (f/g)(x)
f(x)/g(x) Example Use the same functions as
above. Then (f/g)(x) (3x 6)/(-3x2 3x 4)
except for x values that make denominator 0
10
Finding the domains of our new functions

• Finding the domains of these new functions is a
  little complicated.
• The first step is to find the domain of each
  function.
• The second step is to find the intersection of
  the domains. This intersection is the domain of
  the new function.
• Unless the operation is division in which case
  you must also exclude x-values that make the
  denominator zero

11
(No Transcript)
12
Lets do an example Let a) Find (f g)(x) and b)
its domain. (f g)(x)
Step 1 The domain of f(x) is -2,?). The
domain of g(x) is 0,?). Step 2 -2,?) ? 0,?)
0,?). Since this is not a quotient I
dont have to worry about division by zero.
13
(No Transcript)
14
Lets do an example Let a) Find (f - g)(x) and b)
its domain. (f - g)(x)
Step 1 The domain of f(x) is (- ?,?). The
domain of g(x) is (- ?,?). Step 2 (- ?,?) ? (-
?,?) (- ?,?). Since this is not a quotient
I dont have to worry about division by zero.
15
(No Transcript)
16
Example Let a) Find (f g)(x) and b) its
domain. (f g)(x)
Step 1 The domain of f(x) is (- ?,?). The
domain of g(x) is (- ?,?). Step 2 (- ?,?) ? (-
?,?) (- ?,?). Since this is not a quotient
I dont have to worry about division by zero.
17
(No Transcript)
18
Another example

1. Find (f /g)(x) and b) its domain.

Step 1 Domain of f(x) is 0,?). Domain of
g(x) is 10, ?) Step 2 0,?) ? 10, ?)
10, ?) Division so we cant plug in
10 so Domain is (10, ?).
19
Example

• Given f(x) x 5 and g(x) x2 -1, find
  (fg)(x), (f - g)(x), (fg)(x) and (f/g)(x)
• (fg)(x) f(x) g(x) (x 5)(x2 1)
• x2 x - 6
• Domain (fg)(x) (- ?, ?)
• (f g)(x) f(x) g(x) (x 5) (x2 1 )
• x 5 x2 1
• -x2 x 4
• Domain (f g)(x) (- ?, ?)

20
Example

• Given f(x) x 5 and g(x) x2 -1, find (f -
  g)(x), (fg)(x) and (f/g)(x)
• (fg)(x) f(x)g(x) (x 5)(x2 1)
• x3 x 5x2 5
• x3 5x2 x 5
• Domain (fg)(x) (- ?, ?)
• (f/g)(x)f(x)/g(x) (x 5)/(x2 1)
• Domain (f/g)(x) All real except x? 1 or 1
• (- ?, -1) U (-1,1) U (1, ?)

21
Homework

• WS 2-1