Arithmetic operations on functions The Difference Quotient Composition of Functions

1. 2.8 Function Operations
Chapter 2 Graphs and Functions

2. Concepts and Objectives
 Function operations
 Arithmetic operations on functions
 The Difference Quotient
 Composition of Functions and Domain

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    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


5. 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




6. 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
 
  
 

7. The Difference Quotient
 Suppose that point P lies on the graph of , y f x
• Qx+h, fx+h
h
•
Px, fx
0
y = fx
Secant line
and suppose h is a
positive number.

8. The Difference Quotient
 Suppose that point P lies on the graph of , y f x
• Qx+h, fx+h
h
•
Px, fx
0
y = fx
Secant line
and suppose h is a
positive number.
   
 
f x h f x
m
x h x
 

 
With these coordinates,
the slope of the line
joining P and Q is

9. The Difference Quotient
 Suppose that point P lies on the graph of , y f x
• Qx+h, fx+h
h
•
Px, fx
0
y = fx
Secant line
and suppose h is a
positive number.
   
 
f x h f x
m
x h x
 

 
   f x h f x
h
 

With these coordinates,
the slope of the line
joining P and Q is

10. The Difference Quotient
 Suppose that point P lies on the graph of , y f x
• Qx+h, fx+h
h
•
Px, fx
0
y = fx
Secant line
and suppose h is a
positive number.
   
 
f x h f x
m
x h x
 

 
   f x h f x
h
 

This slope is called the
difference quotient and the
line is called a secant line.
With these coordinates,
the slope of the line
joining P and Q is

11. The Difference Quotient (cont.)
 Example: Let . Find the difference
quotient and simplify the expression.
  2
2 3f x x x 

12. The Difference Quotient (cont.)
 Example: Let . Find the difference
quotient and simplify the expression.
There are three pieces of the difference quotient:
fx+h, fx, and h. We already have fx and h, so we
just have to figure out fx+h:
  2
2 3f x x x 
     2
2 3f x h x h x h    

13. The Difference Quotient (cont.)
 Example: Let . Find the difference
quotient and simplify the expression.
There are three pieces of the difference quotient:
fx+h, fx, and h. We already have fx and h, so we
just have to figure out fx+h:
  2
2 3f x x x 
     2
2 3f x h x h x h    
   2 2
2 2 3x xh h x h    
2 2
2 4 2 3 3x xh h x h    

14. The Difference Quotient (cont.)
 Example (cont.): Now we put everything together.
     2 2 2
2 4 2 3 3 2 3x xh h x h x xf x h f x
h h
      

2 2 2
2 4 2 3 3 2 3x xh h x h x x
h
     

2
4 2 3xh h h
h
 

 4 2 3
4 2 3
h x h
x h
h
 
   

15. Composition of Functions
 If f and g are functions, then the composite function, or
composition, of g and f is defined by
 The domain of g ∘ f is the set of all numbers x in the
domain of f such that fx is in the domain of g.
 So, what does this mean?
     g f x g f x

16. Composition (cont.)
 Example: A $40 pair of jeans is on sale for 25% off. If
you purchase the jeans before noon, the store offers an
additional 10% off. What is the final sales price of the
jeans?
We can’t just add 25% and 10% and get 35%. When
it says “additional 10%”, it means 10% off the
discounted price. So, it would be
 
 
25% off: .75 40 $30
10% off: .90 30 $27



17. Evaluating Composite Functions
 Example: Let and .
(a) Find (b) Find
  2 1f x x   
4
1
g x
x


  2f g   3g f 

18. Evaluating Composite Functions
 Example: Let and .
(a) Find (b) Find
(a)
  2 1f x x   
4
1
g x
x


  2f g   3g f 
     2 2f g f g
 
4 4
4
2 1 1
f f f
   
        
 2 4 1 8 1 7    

19. Evaluating Composite Functions
 Example: Let and .
(a) Find (b) Find
(b)
  2 1f x x   
4
1
g x
x


  2f g   3g f 
     3 3g f g f  
      2 3 1 6 1 7g g g       
4 4 1
7 1 8 2
   
  

20. Composites and Domains
 Given that and , find
(a) and its domain
The domain of f is the set of all nonnegative real
number, [0, ∞), so the domain of the composite
function is defined where g ≥ 0, thus
 f x x   4 2g x x 
  f g x
       4 2f g x f g x f x   4 2x 
4 2 0x  
1
2
x  
1
so ,
2
 
  

21. Composites and Domains
 Given that and , find
(b) and its domain
The domain of f is the set of all nonnegative real
number, [0, ∞). Since the domain of g is the set of all
real numbers, the domain of the composite function
is also [0, ∞).
 f x x   4 2g x x 
  g f x
       g f x g f x g x  4 2x 

22. Composites and Domains (cont.)
 Given that and , find
and its domain
 
6
3
f x
x


 
1
g x
x

  f g x
   1
f g x f
x
 
  
 
6
1
3
x


6 6
1 3 1 3x x
x x x
 


6
1 3
x
x



23. Composites and Domains (cont.)
 Given that and , find
The domain of g is all real numbers except 0, and the
domain of f is all real numbers except 3. The expression
for gx, therefore, cannot equal 3:
 
6
3
f x
x


 
1
g x
x

1
3
x

1 3x
1
3
x 
 
1 1
,0 0, ,
3 3
   
      
   

24. 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