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 > 10
-3 <x <23

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 > 10 would be (10,∞)
-3 <x <23 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.
-∞ < x <∞
Use Set Notation:
Use Interval Notation:
(-∞, ∞)

15. How do you express this domain?
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: 4
Name it:
The set of all real numbers greater than or equal to 4.
4 < x <∞
Use Set Notation:
Use Interval Notation:
[4, ∞)

17. Range Range The set of all dependent variables (the y-coordinates)
for which the function is defined

18. What makes a relation a function?
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 y=f(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. Example: 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

31. Viewing and Interpreting Graphs
Recognize that the graph is reasonable.
See all important characteristics of the graph.
Interpret those characteristics.
Recognize grapher failure.