College Algebra
Chapter 2
Functions and Graphs
Section 2.8
Algebra of Functions and
Function Composition
Concepts
1. Perform Operations on Functions
2. Evaluate a Difference Quotient
3. Compose and Decompose Functions
Operations on Functions
Sum
( f  g )( x)  f ( x)  g ( x)
Difference
( f  g )( x)  f ( x)  g ( x)
Product
( f  g )( x)  f ( x)  g ( x)
Quotient
f 
f ( x)
provided that g ( x)  0
  ( x) 
g ( x)
g
Examples 1 – 4:
2
Given f  x   x  3x  1, g  x   8  x , h  x  
,
x 1
evaluate the function, if possible.
2
1. f  3
3. f  a  1
2. g  4 
4. g 10 
Examples 5 – 7:
2
Given f  x   x  3x  1, g  x   8  x , h  x  
,
x 1
evaluate the function, if possible.
2
5.
 f  g  8
7. h  1
6.
 g  h  0 
Example 8:
Evaluate and write the domain in interval notation.
2
s  x 
x
 r  t  x 
t  x  x  4
r  x   x2  3
Example 9:
Evaluate and write the domain in interval notation.
2
s  x 
x
r
  x
t
t  x  x  4
r  x   x2  3
Example 10:
Evaluate and write the domain in interval notation.
2
s  x 
x
 s  t  x 
t  x  x  4
r  x   x2  3
Concepts
1. Perform Operations on Functions
2. Evaluate a Difference Quotient
3. Compose and Decompose Functions
Difference Quotient
The average rate of change between P and Q is the slope
of the secant line and is given by:
y f ( x  h)  f ( x) f ( x  h)  f ( x)
m


x
( x  h)  x
h
The expression
f ( x  h)  f ( x )
h
is called the
difference quotient.
Example 11:
f ( x  h)  f ( x )
Find f ( x) and
for f  x   5 x  2.
h
Example 12:
f ( x  h)  f ( x )
Find f ( x) and
for f  x   x 2  3 x  8.
h
Example 12 continued:
Concepts
1. Perform Operations on Functions
2. Evaluate a Difference Quotient
3. Compose and Decompose Functions
Composition of Functions
The composition of f and g, denoted f
is defined by ( f
g )( x)  f ( g ( x)) .
g
Example 13 continued:
Given f ( x)  x  3 x  1 and g ( x)  5 x  4, find
2
( f  g )( x) 
Example 13:
Given f ( x)  x  3 x  1 and g ( x)  5 x  4, find
2
( g f )( x) 
Example 14:
Given f ( x)  5 x 2  1 and g ( x)  5 x  5, find
( f  g )( x) 
( g f )( x) 
Example 15:
3
Given f  x  
and g ( x)  2 x  8, find
x4
( f g )(9) 
( g f )(6) 
Example 16:
3
Given f  x  
and g ( x)  3 x  2, find ( f g )( x)
x2
and the domain of ( f g )( x).
Example 17:
Given h  x   3 x  7 find two functions f and g
such that h( x)   f g  x  .
Example 18:
Given h  x    x  1  8 find two functions f and g
3
such that h( x)   f g  x  .
Example 19:
Estimate the function values from the graph.
a.
 f  g  3
b.
g
   2 
 f 
Example 19 continued:
c.
g
f 1
d.
f
g  3
e.
f
f  3
Example 20:
A party balloon is being filled with helium. As the
balloon is filling, the radius of the balloon is changing at
the rate of 3 inches per second.
a.
Write a function that represents the radius of the
balloon r(t) after t seconds.
b.
Write a function that expresses the volume of the
balloon V (r) as a function of its radius r.
Example 20 continued:
c.
Evaluate V r  t  and interpret the meaning in
the context of this problem.
Example 20 continued:
d.
Evaluate V r  2  and interpret the meaning in
the context of this problem.