AP Calculus Notes

1
AP Calculus Notes

• Section 1.2
• 9/5/07

2
Objectives

• 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.

3
Key Ideas

• Functions
• Domains and Ranges
• Viewing and Interpreting Graphs
• Even Functions and Odd functions symmetry
• Functions defined in pieces
• The Absolute Value Function
• Composite Functions

4
Functions

• Independent Variable vs. Dependent Variable
• Domain and Range
• Natural Domain
• Boundaries, boundary points, and interval
  notation
• Functions
• Function notation

5
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)
6
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
7
Open vs. Closed Intervals

• The domains and ranges of many real-valued
  functions are intervals or combinations of
  intervals.
• These intervals may be open, closed, or half-open.

8
There are 4 ways to express domains
Graph it
Name it
Use Set Notation
Use Interval Notation
9
What is set notation?
Set notation is what you have used in the past. .
.
For example. . .x gt 10 -3
ltx lt23
10
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 gt 10 would be (10,8)
-3 ltx lt23 would be -3, 23
11
How does set notation compare to interval
notation?
Both are used to indicate sets of numbers
12
Example
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?
13
Is the domain of our example
An open or closed interval?
Open intervals contain no boundary points.
Closed intervals contain their boundary points.
14
The 4 ways to express our domain
Graph it
Name it
The set of all real numbers.
-8 lt x lt8
Use Set Notation
(-8, 8)
Use Interval Notation
15
Example
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?
16
The 4 ways to express our domain
Graph it
Name it
The set of all real numbers greater than or equal
to 4.
4 lt x lt8
Use Set Notation
4, 8)
Use Interval Notation
17
Range
Range
The set of all dependent variables (the
y-coordinates) for which the function is defined
18
Functions
What makes a relation a function?
Consider functions
geometrically

analytically
19
Geometrically speaking. . .
The graph must pass the vertical line test
Are the following functions?
Can you explain why?
20
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
21
Function or not?
D
R
22
Even Odd Functions
By Definition A function yf(x) is an even
function of x if f(-x) f(x) odd function of x
if f(-x) -f(x)
23
With respect to symmetry. . .

• Even functions are symmetric about the y-axis
• Odd functions are symmetric about the origin

24
An example of an even function
25
An example of an odd function
26
Piecewise Functions
Functions that are defined by applying different
formulas to different parts of their domains.
Example
27
Graph it.
28
Absolute Value Function
Absolute Value Functions can also be thought of
as piecewise functions.
Example
29
Composite Functions
30
(No Transcript)
31
Viewing and Interpreting Graphs

• Recognize that the graph is reasonable.
• See all important characteristics of the graph.
• Interpret those characteristics.
• Recognize grapher failure.

32
(No Transcript)