Warm-up:

Homework:

STANDARD 2.3: determine the continuity and end behavior of functions (3-5)

In this section we will…

Determine the continuity or discontinuity of a function.

Identify the end behavior of functions.

Determine whether a function is increasing or decreasing on an interval.

Continuity: A continuous function’s graph can be drawn

without ever lifting up your pencil.

It has no holes or gaps.

All x-values are defined.

Continuous or Not Continuous?

Testing for Continuity:A function is continuous at if it satisfies ALL of the following conditions:

1) the function is defined at ; in other words, ( ) exists.

2) the function approaches the same -valu

x c

c f cAND

y

e on the left and on the right sides of

3) the -value that the function approaches from each side is ( ).

x cAND

yf c

Continuous at x=c?

Types of discontinuities: p 159

Function is undefined at a value but, otherwise, the graph matches up.

Graph has a “hole”.

Types of discontinuities: p 159

Graph stops at one y-value, then “jumps” to a different y-value for the same x-value.

Common in piece-wise functions.

Types of discontinuities: p 159

A major disruption in the graph.

As graph approaches the domain restriction, the graph will shoot towards either positive or negative infinity.

1. Continuous or Not Continuous

2. If not continuous, Type of Discontinuity

3. Which test it failed (only need one)

4. Domain and Range

For the following examples, find the following

Continuity on an interval: A function is continuous on an interval iff it

is continuous at each number in the interval.

Increasing and decreasing functions:

A f unction is said to be increasing on an interval, , iff

f or every and contained on , ( ) ( ), wherever .

A f unction is said to be decreasing on an interval, , iff

f or every and containe

I

a b I f a f b a b

I

a b

d on , ( ) ( ), wherever .

A f unction is said to be constant on an interval, , iff

f or every and contained on , ( ) ( ), wherever .

I f a f b a b

I

a b I f a f b a b

Ahhhh…Huh? Increasing means uphill left to right.

Decreasing means downhill left to right.

Constant means a flat or horizontal line left to right.

Behavior over an interval and interval notation:

Graph these on your calculator:

P 166 #26, 28, 30

Determine the intervals where the functions are increasing or decreasing.

Write the intervals in interval notation and in in terms of x.

Answers:

26.

28.

30.

End Behavior: What will the function be doing at the

outermost reaches of its domain and range?

Check these out!

1( )

3

3, 2( )

3 2, 2

f xx

xf x

x x

Given the following function, determine its continuity, behavior over its domain and end behavior.

TOD:

5 2( ) 5 3f x x x

Homework:

HW 2.3: P 166 #13 – 31 odd, 39

You will need a graphing calculator.