Math 95
10.6 "Algebra and Composition of Functions"
Objectives:
*
Add, subtract, multiply, and divide functions.
*
Find the composition of functions and use graphs to evaluate functions.
*
The identity function and the di¤erence quotient.
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]
[Product]
[Di¤erence]
[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
c) f g
d)
g
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. 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 composite function, denoted by f
g ("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
Page: 1
g:
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
10.6
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
4: Find:
g) (9)
c) (g f ) ( 2)
b) (f
g) (x)
d) (f
g) ( 2)
Example 3: (Composite functions)
Let f (x) = 3x
2 and g (x) = x2 + x + 1: Find:
a) (f
g) (2)
b) (g f ) (2)
c) (f
g) (x)
d) (g f ) (x)
Page: 2
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
10.6
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 5: (Using graphs to evaluate functions)
Refer to the graphs of functions f (red) and g (blue) to …nd each of the following
y
a) (f + g) ( 4)
4
b) (f g) (2)
c) (f
2
g) ( 3)
d) (f g) (3)
f
( 2)
e)
g
-6
-4
-2
2
4
-2
6
x
f) (g f ) (3)
-4
The Identity Function
The identity functions is de…ned by the equation I (x) = x. Under this function, the value that corresponds
to any real number x is x itself. If f is any function, the composition of f with the identity function is the function
f:
The Di¤erence Quotient
An important function in calculus, called the di¤erence quotient, represents the slope of a line that passes through two
given points on the graph of a function. The di¤erence quotient is de…ned as follows:
Page: 3
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
10.6
Example 6: (Finding the di¤erence quotient)
Find the di¤erence quotient of the following functions.
a) f (x) = 3x
5
b) f (x) = x2
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4x + 3
Notes by Bibiana Lopez