Objectives:
1. To find the difference
quotient of a function
2. To add, subtract,
multiply, and divide
functions and find
their domain
3. To find the
composition of two
functions and find its
domain
•
•
•
•
•
Assignment:
P. 89: 5-12 S
P. 90: 35-42 S
P. 90: 47-54 S
Homework
Supplement
Read: P. 93-98
Difference Quotient
Pascal’s Triangle
Function Composition
Function
Decomposition
You will be able to find the
difference quotient of a function
•
•
Eventually, you will learn
to take the limit of this
quotient as h
approaches 0. That’s
known as the
derivative of the
function. But for now,
we’re just going to
evaluate the thing.
•
It’s just the slope of a secant line!
•
1
1
1
2
3
1
1
1
4
1
1
3
6
4
1
1. Evaluate f (x + h)
separately.
2. Evaluate f (x + h) first,
then find the
difference quotient.
3. Use Pascal’s Triangle
to help evaluate
(x + h)n.
Coefficients:
 x  h  1
0
1
Degree -> (Starting at 0)
1
1
1
2
3
1
4
 x  h
1
1
4
2
 x 2  2 xh  h 2
 x  h   x3  3x 2 h  3xh 2  h3
3
1
3
6
 x  h
1
 xh
1
 x  h
4
 x 4  4 x3h  6 x 2 h 2  4 xh3  h 4
Coefficients:
 x  h  1
0
1
Degree -> (Starting at 0)
1
1
1
2
3
1
4
 x  h
1
1
4
2
 x 2  2 xh  h 2
 x  h   x3  3x 2 h  3xh 2  h3
3
1
3
6
 x  h
1
 xh
1
 x  h
4
 x 4  4 x3h  6 x 2 h 2  4 xh3  h 4
•
You will be able to add,
subtract, multiply, and divide
functions and find their domain
A function is a
relation in which
each input has
exactly one output.
• A function is a
dependent relation
• Output depends on
the input
Domain:
The set of all
input values
Range:
The set of all
output values
Remember adding, subtracting, multiplying, and
dividing numbers? Well, we can do all of those
things with functions to make new functions!
When +/−
functions:
When 
functions:
When 
functions:
Just collect
like terms
Use the
distributive
property
Write functions
as a quotient
and simplify
Remember adding, subtracting, multiplying, and
dividing numbers? Well, we can do all of those
things with functions to make new functions!
Remember adding, subtracting, multiplying, and
dividing numbers? Well, we can do all of those
things with functions to make new functions!
Sum:
( f  g )( x)  f ( x)  g ( x)
Difference:
( f  g )( x)  f ( x)  g ( x)
Product:
( fg )( x)  f ( x)  g ( x)
Quotient:
f 
f ( x)
, g ( x)  0
  ( x) 
g ( x)
g
The domain of h consists of all x-values that are
common to domains of BOTH f and g.
Domain
of f
Domain
of g
h 



f g
f g
f g
f g
Domain of h
Except where g(x) =0
The domain of h consists of all x-values that are
common to domains of BOTH f and g.
f
h 



g
f g
f g
f g
f g
Domain of h
Except where g(x) =0
The domain of h consists of all x-values that are
common to domains of BOTH f and g.
f
h 



g
f g
f g
f g
f g
Domain of h
Except where g(x) =0
•
•
•
Just like with real numbers, function addition
and multiplication are commutative. In other
words, the order doesn’t matter.
f ( x)  g ( x)  g ( x)  f ( x)
f ( x)  g ( x)  g ( x)  f ( x)
I wonder if other properties hold as well…
A small company sells computer printers over
the Internet. The company’s total monthly
revenue (R) and costs (C) are modeled by the
functions R(x) = 120x and C(x) = 2500 + 75x,
where x is the number of printers sold.
1. Find R(x) – C(x)
2. Explain what this difference means
You will be able to find the composition of
two functions and find its domain
Function composition happens when we take a
whole function and substitute it in for x in
another function.
h( x)  g  f ( x) 
Substitute f(x) in for x in g(x)
– The “interior” function gets substituted in for x in
the “exterior” function
Function composition happens when we take a
whole function and substitute it in for x in
another function.
h( x)   g f  ( x)
Substitute f(x) in for x in g(x)
– Read as “g composed with f ”
In function
composition, the
domain of h is the
set of x in the
domain of f such
that f (x) is in the
domain of g.
h( x)  g  f ( x) 
In other words, we can
only use x values
that are in the
domain of the
interior function
whose output is also
in the domain of the
exterior function.
h( x)  g  f ( x) 
In other other words,
the domain of h is
the domain of the
interior function f as
long as its range is
completely within
the domain of the
exterior function g.
h( x)  g  f ( x) 
•
•
•
•
•
•
Rather than composing
two functions to make
one function,
sometimes we want to
take a single function
h(x) and find two
functions f (x) and g(x)
whose composition
yields h(x). This is
called decomposition.
g ( x)  2 x  3
h( x)  (2 x  3) 2
f ( x)  x 2
h( x)   f g  ( x)
Write the function below as a composition of
two functions.
8 x
h( x ) 
5
3
•
f ( x)  2 x  3
g ( x)  x 2
h( x )  3 x  1
•
Objectives:
1. To find the difference
quotient of a function
2. To add, subtract,
multiply, and divide
functions and find
their domain
3. To find the
composition of two
functions and find its
domain
•
•
•
•
•
Assignment:
P. 89: 5-12 S
P. 90: 35-42 S
P. 90: 47-54 S
Homework
Supplement
Read: P. 93-98