Download Composite Functions

Avon High School
Section: 2.6
ACE COLLEGE ALGEBRA II - NOTES
Combinations of Functions: Composite Functions
Mr. Record: Room ALC-129
Day 1 of 1
The Domain of a Function
Consider the following functions that model the data in the table
to the left.
B( x)  7.4 x 2  15x  4046
D( x)  3.5x 2  20 x  2405
Number of births, B(x), in
thousands, x years after 2000
Number of deaths, D(x), in
thousands, x years after 2000
Because the years in this data model extend from 2000 through
2005, both the domain of B(x) and D(x) are considered to be
x | x  0,1, 2,3, 4,5 . Functions that model data often have their
domains explicitly given with the function’s equations. However, most functions are represented by only by an
1
or g ( x)  x  2 . What are their domains?
equation. Consider f ( x) 
x 3
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real
numbers for which the value of f (x) is a real number Exclude from a function’s domain real
numbers that cause division by zero and real numbers that result in a square root of a negative
number
Example 1
Finding the Domain of a Function
Find the domain of each function.
5x
a. f ( x)  x 2  3x  17
b. g ( x)  2
x  49
c. h( x)  9 x  27
The Algebra of Functions
Sum, Difference, Product and Quotients of Functions
Let f and g be two functions. The sum f + g, difference f  g , the product f  g , and the quotient
f
are functions whose domains are the set of real numbers common to the domains of f and g
g
D
f
Dg  , defined as follows:
1. Sum: ( f  g )( x)  f ( x)  g ( x)
3. Product: ( f  g )( x)  f ( x)  g ( x)
2. Difference: ( f  g )( x)  f ( x)  g ( x)
f 
f ( x)
, provided g ( x)  0.
4. Quotient:   ( x) 
g ( x)
g
Example 2
Combining Functions
Let f ( x)  x  5 and g ( x)  x 2  1 . Find each of the following functions:
a. ( f  g )( x )
b. ( f  g )( x )
f 
d.   ( x)
g
c. ( f  g )( x)
Investigation: Use a graphing calculator to sketch the function y  x  3  x  2 in a [-3,10,1] by
[0,8,1] viewing rectangle. Does this reinforce what we learned about the domain of a combination of
two functions?
Composite Functions
The Composition of Functions
The composition of the function f with g is denoted by f g and is defined by the equation
( f g )( x)  f ( g ( x)) .
The domain of the composite function f ○ g is the set of all x such that
1. x is in the domain of g and
2. g(x) is in the domain of f.
Example 3
a.
f
g  ( x)
Forming Composite Functions
Given f ( x)  5x  6 and g ( x)  2 x 2  x  1 , find each of the following:
b.
g
f  ( x)
c.
f
g  (1)
We must be careful in determining the domain for a composite function.
Excluding Values from the Domain of (f ○g )(x) = f (g (x))
The following values must be excluded from the input x:
 If x is not in the domain of g, it must not be in the domain of f g .
 Any x for which g ( x ) is not in the domain of f must not be in the domain of f g .
Example 4
Given f ( x) 
f
a.
Forming a Composite Function and Finding Its Domain
4
1
and g ( x)  , find each of the following:
x2
x
g  ( x)
b. the domain of  f g  ( x)
Decomposing Functions
Example 5
Writing a Function as a Composition
Express h( x) as a composition of two functions where h( x)  x 2  5