Arithmetic operations on functions

1. 2.8A Function Operations
Chapter 2 Graphs and Functions

2. Concepts and Objectives
 Function operations
 Arithmetic operations on functions

3. Operations on Functions
 Given two functions f and g, then for all values of x for
which both fx and gx are defined, we can also define
the following:
 Sum
 Difference
 Product
 Quotient
      f g x f x g x  
      f g x f x g x  
      fg x f x g x 
 
 
 
 , 0
f xf
x g x
g g x
 
  
 

4. Operations on Functions (cont.)
 Example: Let and . Find each
of the following:
a)
b)
c)
d)
  2
1f x x    3 5g x x 
  1f g
  3f g 
  5fg
 0
f
g
 
 
 

5. Operations on Functions (cont.)
 Example: Let and . Find each
of the following:
a)
b)
c)
d)
  2
1f x x    3 5g x x 
  1f g    1 1gf   2
51 11 3   02 18  
  3f g 
  5fg
 0
f
g
 
 
 

6. Operations on Functions (cont.)
 Example: Let and . Find each
of the following:
a)
b)
c)
d)
  2
1f x x    3 5g x x 
  1f g    1 1gf   2
51 11 3   02 18  
  3f g     2
3 53 31        410 14  
  5fg
 0
f
g
 
 
 

7. Operations on Functions (cont.)
 Example: Let and . Find each
of the following:
a)
b)
c)
d)
  2
1f x x    3 5g x x 
  1f g    1 1gf   2
51 11 3   02 18  
  3f g     2
3 53 31        410 14  
  5fg    2
3 5 55 1       02026 52 
 0
f
g
 
 
 

8. Operations on Functions (cont.)
 Example: Let and . Find each
of the following:
a)
b)
c)
d)
  2
1f x x    3 5g x x 
  1f g    1 1gf   2
51 11 3   02 18  
  3f g     2
3 53 31        410 14  
  5fg    2
3 5 55 1       02026 52 
 0
f
g
 
 
   
2
5
0 1
3 0



5
1


9. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
a)
b)
c)
d)
  8 9f x x    2 1g x x 
  f g x
  f g x
  fg x
 
f
x
g
 
 
 

10. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
a)
b)
c)
d)
  8 9f x x    2 1g x x 
  f g x 8 9 2 1x x   
  f g x
  fg x
 
f
x
g
 
 
 

11. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
a)
b)
c)
d)
  8 9f x x    2 1g x x 
  f g x 8 9 2 1x x   
  f g x 8 9 2 1x x   
  fg x
 
f
x
g
 
 
 

12. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
a)
b)
c)
d)
  8 9f x x    2 1g x x 
  f g x 8 9 2 1x x   
  f g x 8 9 2 1x x   
  fg x  8 9 2 1x x  
 
f
x
g
 
 
 

13. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
a)
b)
c)
d)
  8 9f x x    2 1g x x 
  f g x 8 9 2 1x x   
  f g x 8 9 2 1x x   
  fg x  8 9 2 1x x  
 
f
x
g
 
 
 
8 9
2 1
x
x




14. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
  8 9f x x    2 1g x x 

15. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
 There are two cases that need restrictions: taking the
square root of a negative number and dividing by zero.
  8 9f x x    2 1g x x 

16. Operations on Functions (cont.)
 Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
 There are two cases that need restrictions: taking the
square root of a negative number and dividing by zero.
 We address these by making sure the inside of gx > 0:
  8 9f x x    2 1g x x 
2 1 0
2 1
1
2
x
x
x
 


So the domain must be
1 1
or ,
2 2
x
 
  
 

17. Classwork
 2.8 Assignment (College Algebra)
 2.8 – pg. 282: 2-14 (even); 2.7 – pg. 271: 24-36
(even); 2.6 – pg. 257: 48-52, 56 (even)
 Classwork Check 2.8
 Quiz 2.7