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 RangeNatural Domain

Boundaries, boundary points, and interval notation

FunctionsFunction 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)

The set of all non-restricted x-valuesNatural Domain

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.

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.

Use Set Notation:Use Set Notation:

Name it:Name it:

There are 4 ways to express domains:

Graph itGraph it:

Use Interval 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.

For example: x > 10 would be (10,∞)

-3 <x <23 would be [-3, 23]

( or ) indicate open boundaries

[ or ] indicate closed boundaries

How does set notation compare to interval notation?

Both are used to indicate sets of numbers

y x 2 8

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.

Are there any restrictions on x ?

Example:

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.

Use Set Notation:Use Set Notation:

Name it:Name it:

The 4 ways to express our domain:

Graph itGraph it:

Use Interval Notation:Use Interval Notation:

0

The set of all real numbers.

-∞ < x <∞

(-∞, ∞)

y x 2 8

Therefore, natural domain is the set of all x values for which 2x – 8 0.

Are there any restrictions on x ?

You cannot have a negative radicand.

Example:

How do you express this domain?

Use Set Notation:Use Set Notation:

Name it:Name it:

The 4 ways to express our domain:

Graph itGraph it:

Use Interval Notation:Use Interval Notation:

4

The set of all real numbers greater than or equal to 4.

4 < x <∞

[4, ∞)

Range

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

Range

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 uniqueunique element in R to each element in D.

DR

Function or not?

D

R

Even & Odd Functions

By Definition:

A function y=f(x) is an

eveneven function of x if ff(-x)= f(x)(-x)= f(x)

oddodd function of x if f(-x) = -f(x)f(-x) = -f(x)

With respect to symmetry. . .

Even functions are symmetric about Even functions are symmetric about the y-axisthe y-axis

Odd functions are symmetric about the Odd functions are symmetric about the originorigin

An example of an even function:

y x 2

f x = x2

An example of an odd function:

y x 3

6

4

2

-2

-4

-6

-5 5

f x = x3

Piecewise FunctionsFunctions that are defined by applying different formulas to different parts of their domains.

f x

x

x

x

( )

RS|T|

1 0

0 0

1 0

if

if

if

ExamplExample:e:

Graph it.4

3

2

1

-1

-2

-3

-4 -2 2 4

Absolute Value Function

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

Example:

y x

y f xx x

x x

RST( ) if

if

0

0

Composite Functions

f g f g x ( ( ))

g f g f x ( ( ))

Viewing and Interpreting Graphs

Recognize that the graph is reasonable.

See all important characteristics of the graph.

Interpret those characteristics. Recognize grapher failure.