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Haberman / Kling MTH 111c
Section I: Sets and Functions
Module 4: The Algebra of Functions
We can use the four basic arithmetic operations (addition, subtraction, multiplication, and
division) to create new functions from old ones.
DEFINITION: If f and g are functions and x represents a value in both of their
domains, then we can define the following four functions:
Sum-of-Functions: ( f + g ) ( ) x = f ( ) x + g x ( )
Difference-of-Functions: ( f โ g ) ( ) x = f ( ) x โ g x ( )
Product-of-Functions: ( f โ
g ) ( ) x = f ( ) x โ
g x ( )
Quotient-of-Functions:
f f x x g x g g x
(it is important to note that
g x ( ) โ 0 since this would
cause division by zero, and
division by zero is undefined)
EXAMPLE: Suppose that the function represents the total number of female
students enrolled at PCC t years after 1990 and that represents the
total number of male students enrolled at PCC t years after 1990. Write an
expression that represents the total number of students enrolled at PCC t
years after 1990.
s = f ( ) t
s = m ( t )
SOLUTION : represents the total number of students enrolled at PCC t years
after 1990.
s = ( f + m ) ( ) t
EXAMPLE: Let (^) h x ( ) = 5 โ 7 x and 2
k x x
.
a. Find and simplify the rule for ( h + k ) ( ) x.
b. Find and simplify the rule for ( h k โ
) ( ) x.
c. Evaluate ( h^ + k ) (2).
d. Evaluate (1)
h
k
.
e. Evaluate ( k โ h ) ( 3)โ.
SOLUTIONS:
a.
2
2
2 2
2
2 2
2
2
3 2
2
h k x h x k x
x x
x x x x
x x
x x
x x
x
x x x
x
b.
2
2
2
h k x x x
x
x
x
x
e.
2
k โ h โ = k โ โ h โ
โ +^ โ โ
EXAMPLE: Given the graphs of y = f ( x ) and y = g ( x ) in Figures 1 and 2, respectively,
graph y = ( f + g ) ( ) x in Figure 3.
Figure 1: Graph of y = f ( ) x. Figure 2: Graph of y = g x ( ).
SOLUTION :
To graph , choose an
input value and add the corresponding
output values. For instance,
y = ( f + g ) ( ) x
f (4) = 4
and , so , while
since
g (4) = 1 ( f + g ) (4) = 5
( f^ +^ g ) ( 6)โ^ = โ^2 f (^ โ^6 )=โ^1
and g ( 6)โ = โ 1.
Figure 3: Graph of y = ( f + g ) ( ) x.
Try this one yourself and check your answer.
If f ( ) x = 2 x โ 1 and g x ( ) = โ x + 3 , find
a. ( f + g ) ( ) x c. ( f โ
g ) ( ) x
b. ( f โ g ) ( ) x d. ( )
f x g
SOLUTIONS:
a. Click here for solution
b. Click here for solution
c. Click here for solution
d.
for
f x x g x
โ โ โ^ +
x โ (it is important to note that^ x^ โ ^3 since this would cause
division by zero, and division by zero is undefined)
Try this one yourself and check your answer.
Fill in the missing parts of the table below.
x f^ ( x ) g^ ( x ) ( f^ +^ g^ ) ( ) x ( f^ โ^ g^ ) ( ) x ( f^ โ
^ g^ ) ( ) x ( f^ /^ g^ )(^ x )^ ( )
2 f ( x )
CLICK HERE
FOR SOLUTION
