The document covers function operations in mathematics, explaining how to perform basic operations like addition, subtraction, multiplication, and division on functions. It includes examples of finding sums and differences of functions, determining their domains and ranges, and emphasizes the importance of combining like terms and simplifying expressions. Additionally, it provides resources for further learning and practice assignments.

1. Math 30-1 1
Ch 10. Function Operations
What is meant by operations in mathematics?
or

2. Math 30-1 2

3. Math 30-1 3
Have I seen operations on functions before??
2
6
y x x
  
2
( )
( )
( ) 6
f x x
g x x
h x



( ) ( ) ( )
y f x g x h x
  
  
3 2
y x x
  
 
 
( ) 3
( ) 2
r x x
s x x
 
 
( ) ( )
y r x s x
 
quadratic
linear
constant
linear
linear
New functions can be formed by performing
operations with functions.

4. Math 30-1 4
Basic Function Operations
• Sum
• Difference
• Product
• Quotient
Restrictions or NPV’s
 ( ) ( ) ( )
f g x f x g x



( ) and ( )
f x g x are functions that exist and are defined over a domain.
 ( ) ( ) ( )
f g x f x g x



 ( ) ( ) ( )
f g x f x g x

 
( )
( ) , ( ) 0
( )
f f x
x g x
g g x
 
 
 
 
Linear Quadratic Radical
linear
Rational
monomial, binomial
Absolute Value
first degree
Cubic Exponential Logarithmic

5. Math 30-1 5
10.1 Sums and Differences of Functions
The sum f + g
      
x
g
x
f
x
g
f 


To find the sum of two functions, add them together.
Simplify by combining like terms.
Algebraically
( ) ( )
f x g x
 
2
3 4 4
x x
 

( )
f x  2
2 3
x  ( )
g x  2
4 1
x x
 

New function

6. Math 30-1 6
      
x
g
x
f
x
g
f 

 Graphing with technology
  2
( ) 3 4 4
f g x x x
   
( )
f x  2
2 3
x 
( )
g x  2
4 1
x x
 

7. Math 30-1 7
The difference f - g
      
x
g
x
f
x
g
f 


To find the difference between two functions, subtract
the second from the first. Simplify by combining like
terms.
Algebraically
( ) ( )
f x g x
 
2 2
2 3 4 1
x x x
    
( )
f x  2
2 3
x  ( )
g x  2
4 1
x x
 
( )

2
4 2
x x
   New function

8. Math 30-1 8
( )
f x  2
2 3
x  ( )
g x  2
4 1
x x
    2
( ) 4 2
f g x x x
   
Sketch the graph using technology and determine the domain and range.

9. Math 30-1 9
(f + g - h)(x)
Given f(x) = 2x + 6 , g(x) = -4x - 3, and write an
expression in simplest form and determine domain and range.
( ) 2
h x x
 
= f(x) + g(x) - h(x)
= (2x + 6) + (-4x - 3) -
= -2x + 3 -
2
x 
2
x 
How does the domain of each individual function affect the domain of the new function?
 
| 2,
x x x
 
 
7,
y y
 
  
2
f g h
  

10. Math 30-1 10
TRUE OR FALSE?
     
h x f x g x
 
The domain of h(x) is the sum of the domains of f(x) and g(x)
The range of h(x) is the sum of the ranges of f(x) and g(x)
  2
4
f x x
   
g x x

 
|
x x  
| 0,
x x x
 
The domain of h(x) is the overlap of the domains of f(x) and g(x)
Determine the range graphically or use substitution.

11. Math 30-1 11
Sketch the Graph of the sum f + g
Consider the functions
( ) ( ) ( )
h x f x g x
 
( )
f x x
 ( ) 3
g x 
What part of the graphs would be added?
x-coordinates? y-coordinates?
( ) 3
h x x
 
Domain
Range

12. Math 30-1 12
10.1 Sums and Differences
http://www.psja.k12.tx.us/~abcron/
GeoGebraMenu/GeogebraFiles/arithmetic/
add_funct.html
http://www.psja.k12.tx.us/~abcron/
GeoGebraMenu/GeogebraFiles/arithmetic/
sub_funct.html
Investigating the Graphs of Sums
and Differences with Technology

13. Math 30-1 13
( ) ( ) ( )
h x f x g x
 
( )
f x x
 ( ) 2
g x 
Sketch a graph of the difference of f(x) and g(x).
( ) 2
h x x
 
Use your knowledge of transformations to verify the graph.
Domain
Range

14. Math 30-1 14
Consider
2
( ) sin
f x x
 2
( ) cos
g x x
 ( ) ( ) ( )
h x f x g x
 
Domain: Domain:
Range: Range:
x
 
0 ( ) 1
f x
 
x
 
0 ( ) 1
g x
 
Is the Domain of ( ) ( ) ( )
h x f x g x
  the sum of the two domains?
Is the Range of ( ) ( ) ( )
h x f x g x
  the sum of the two ranges?
x
 
1

15. Math 30-1 15
Consider and .
1
( )
3
f x
x


( )
11
x
g x
x


Express the sum h(x) = (f + g)(x) in simplest form
and state the domain and range.
( ) ( ) ( )
h x f x g x
 
1
( )
3 11
x
h x
x x
 
 
   
  
1 11 3
( )
3 11
x x x
h x
x x
  

 
  
2
4 11
( )
3 11
x x
h x
x x
 

 
 
| 3,11,
x x x
 
 
|
y y 

16. Math 30-1 16
In Summary
To sketch the graph of a sum of two functions given their
graphs, ______ the ____coordinates at each x-value.
add y
To sketch the graph of a difference of two functions given
their graphs, __________ the ____coordinates at each x-value.
subtract y
Domain and Range
The domain of f(x) + g(x) is the overlap (intersection) of the
domains of the individual functions.
The range of f(x) + g(x) is determined from the graph of the
sum or difference.

17. Math 30-1 17
Assignment
Page 483
1a,c,d, 2c,d, 4a,b, 5, 6, 7, 8, 9c,d, 10(new), 11a,b, 13, 14,
C1, C2

18. Math 30-1 18
http://www.intmath.com/functions-and-
graphs/2a-domain-and-range.php
Domain and Range Review
http://www.mathsisfun.com/sets/functions-
operations.html
http://www.wtamu.edu/academic/anns/mps/
math/mathlab/col_algebra/
col_alg_tut30b_operations.htm
Operations with Functions
Operations with Functions