Download Add, Subtract, Multiply, and Divide Functions Find the Composition

Math 35 (Spring ’09)
9.1 "Algebra and Composition of Functions"
Objectives:
*
Add, subtract, multiply, and divide functions
*
Find the composition of functions
*
Use graphs to evaluate functions
Add, Subtract, Multiply, and Divide Functions
Operations on Functions:
If the domains and ranges of functions f and g are subsets of the real numbers, then:
[Sum]
[Di¤erence]
[Product]
[Quotient g (x) 6= 0 ]
The domain of each of these functions is the set of real numbers x that are in the domain of both f and g.
Example 1: (Sum/Di¤erence/Product/Quotient)
Let f (x) = 4x2 9 and g (x) = 2x 3. Find the following and simplify:
a)
f +g
b)
f g
c)
f g
d)
f
g
Find the Composition of Functions
We have seen that a function can be represented by a machine: We put in a number from the domain, and a number
from the range comes out. For example:
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Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
9.1
Suppose that y = f (x ) and y = g (x) de…ne two functions. Any number x in the domain of g will produce the corresponding value g (x) in the range of g. If g (x) is in the domain of function f , then g (x) can be substituted into f , and a corresponding
value f (g (x)) will be determined. This two-step process de…nes a new function called a
,
denoted by
("f composed with g" or "the composition of f and g").
The function machines can illustrate the composition f g. When we put a number into the function g, a value g (x)
comes out. The value g (x) then goes into function f , which transforms g (x) into f (g (x)) ("f of g of x"). If the function
machines for g and f were connected to make a single machine, that machine would be named f g:
Composite Functions:
The composite function f
g is de…ned by:
Example 2: (Composite functions)
Let
f (x) = 2x + 1 and g (x) = x
a)
(f
c)
(g f ) ( 2)
4: Find:
g) (9)
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b)
(f
g) (x)
d)
(f
g) ( 2)
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
9.1
Example 3: (Composite functions)
Let
f (x) = 3x
a)
(f
g) (2)
c)
(f
g) (x)
2 and g (x) = x2 + x + 1: Find:
b)
d)
(g f ) (2)
(g f ) (x)
Example 4: (Composite functions)
If f (x) = x + 1 and g (x) = 2x
5; show that (f
g) (x) 6= (g f ) (x)
Use Graphs to Evaluate Functions
Example 6: (Using graphs to evaluate functions)
Refer to the graphs of functions f (red) and g (blue) to …nd each of the following
a)
(f + g) ( 4)
b)
(f g) (2)
c)
(f
d)
(f g) (3)
f
( 2)
g
e)
f)
y
4
2
g) ( 3)
(g f ) (3)
6
-6
-4
-2
2
-2
4
6
x
-4
-6
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Notes by Bibiana Lopez