AP Calculus Notes Section 1.2 9/5/07

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.

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

Functions 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 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: 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: The set of all real numbers. -∞ < x <∞ Use Set Notation: Use Interval Notation: (-∞, ∞)

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?

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, ∞)

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

What makes a relation a function? 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. . . Even functions are symmetric about the y-axis Odd functions are symmetric about the origin

An example of an even function:

An example of an odd function:

Example: Piecewise Functions Functions that are defined by applying different formulas to different parts of their domains. Example:

Graph it.

Absolute Value Function Absolute Value Functions can also be thought of as piecewise functions. Example:

Composite Functions

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