AP Calculus Notes
Section 1.2
9/5/07
Objectives
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Students will be able to identify the domain
and range of a function using its graph or
equation.
Students will be able to recognize even
functions and odd functions using equations
and graphs.
Students will be able to interpret and find
formulas for piecewise defined functions.
Students will be able to write and evaluate
compositions of two functions.
Key Ideas
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Functions
Domains and Ranges
Viewing and Interpreting Graphs
Even Functions and Odd functions –
symmetry
Functions defined in pieces
The Absolute Value Function
Composite Functions
Functions
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Independent Variable vs. Dependent
Variable
Domain and Range
 Natural Domain
 Boundaries, boundary points,
and interval notation
Functions
 Function notation
Independent Variables
vs.
Dependent Variables
The independent variable is the
first coordinate in the ordered
pair (the x values)
The dependent variable is the
second coordinate in the
ordered pair (the y values)
Domain
The set of all independent variables
(x-coordinates)
If the domain of a function is not stated
explicitly, then assume it to be the
largest set of real x-values for which the
equation gives real y-values.
Any exclusions must be specifically
stated.
Natural Domain
The set of all non-restricted x-values
Open vs. Closed Intervals
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The domains and ranges of many
real-valued functions are intervals
or combinations of intervals.
These intervals may be open,
closed, or half-open.
There are 4 ways to express
domains:
Graph it:
Name it:
Use Set Notation:
Use Interval Notation:
What is set notation?
Set notation is what you have used
in the past. . .
For example. . .x > 10
-3 <x <23
What is interval notation?
Interval notation uses ( ,[, ), or ] to
denote the set of numbers to which
you refer.
( or ) indicate open boundaries
[ or ] indicate closed boundaries
For example: x > 10 would be (10,∞)
-3 <x <23 would be [-3, 23]
How does set notation compare to
interval notation?
Both are used to indicate sets of
numbers
Example:
y  2x  8
Are there any restrictions on x ?
Because there is no restriction on
the possible values that may be
used for x, the natural domain is
the set of all real numbers.
How do you express this domain?
Is the domain of our example
An open or closed interval?
Open intervals contain no
boundary points.
Closed intervals contain
their boundary points.
The 4 ways to express our domain:
Graph it:
Name it:
0
The set of all real numbers.
Use Set Notation:
-∞ < x <∞
Use Interval Notation:
(-∞, ∞)
Example:
y  2x  8
Are there any restrictions on x ?
You cannot have a negative radicand.
Therefore, natural domain is the set
of all x values for which 2x – 8  0.
How do you express this
domain?
The 4 ways to express our domain:
Graph it:
Name it:
4
The set of all real numbers
greater than or equal to 4.
Use Set Notation:
4 < x <∞
Use Interval Notation:
[4, ∞)
Range
Range
The set of all dependent
variables (the y-coordinates)
for which the function is defined
Functions
What makes a relation a
function?
Consider functions
geometrically
&
analytically
Geometrically speaking. . .
The graph must pass the vertical line
test:
Are the following functions?
Can you explain
why?
Analytically. . .
By Definition:
A function from a set D to a set
R is a rule that assigns a unique
element in R to each element in
D.
D
R
Function or not?
D
R
Even & Odd Functions
By Definition:
A function y=f(x) is an
even function of x if f(-x)= f(x)
odd function of x if f(-x) = -f(x)
With respect to symmetry. . .
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Even functions are symmetric about
the y-axis
Odd functions are symmetric about the
origin
An example of an even function:
yx
2
fx = x2
An example of an odd function:
yx
fx =
3
6
x3
4
2
-5
5
-2
-4
-6
Piecewise Functions
Functions that are defined by
applying different formulas to
different parts of their domains.
Example:
1 if x  0
R
|
f ( x)  S
0 if x  0
|T1 if x  0
Graph it.
4
3
2
1
-4
-2
2
-1
-2
-3
4
Absolute Value Function
Absolute Value Functions can also
be thought of as piecewise
functions.
Example:
y x
 x if x  0
R
y  f ( x)  S
Tx if x  0
Composite Functions
f  g  f ( g( x ))
g  f  g( f ( x ))
Viewing and Interpreting Graphs
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Recognize that the graph is
reasonable.
See all important characteristics of
the graph.
Interpret those characteristics.
Recognize grapher failure.