Warm up Graph the function and its inverse:

Find for the relation . Then state whether is a function

12 xy

)(1 xf

xxxf 32)(

)(1 xf

Lesson 3-5 Continuity and End BehaviorObjective: To determine whether a function is continuous or discontinuousTo identify the end behavior of functionsTo determine whether a function is increasing or decreasing on an interval

Discontinuity – a break in the graph.

There are different types of discontinuity:Infinite Discontinuity :|f(x)| becomes

greater & greater as the graph approaches a given value.

Jump Discontinuity- the graph stops at a given value of the domain and then

begins again at a different range value for the same value of the domain.

Point Discontinuity- there is a value of the domain where the function is undefined.

Everywhere Discontinuous- impossible to graph in the real number system.

ex:

irrationalisxifrationalisxif

xf11

)(

Which of the following does not display jump continuity?

A

DC

B

Continuous passes through all the points of the

graph without a break.

Linear and quadratic functions are continuous.

Continuity Test – must satisfy all 3 conditionsA function is continuous at x = c if:1. the function is defined at c (f(c)

exists)2. the function approaches the

same y-value on the left and on the right sides of x=c.

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

ExampleDetermine whether each function is

continuous at the given x-value.1.

2.

3.

1;73 2 xxxy

2;24)(

2

xxxxf

22

41

2)(

2

2

x

xifx

xifxxf

Continuity

You can also look at continuity over a given interval of the graph instead of the whole graph.

Continuity on an interval: a function f(x) is continuous on an interval if & only if it is continuous at each number x on the interval.

Example The U.S Postal Service offers insurance for its

express mail. For a package valued at $500 or less, insurance is included in the $11.75 fee. For $500.01 to $5000, it costs an additional $0.95 per $100 of value. Show the step graph that represents this

situation.Use the continuity test to show that the step

function is discontinuous.Explain why a continuous function would not be

appropriate to model express mail rates.

Warm upDetermine whether each function is

continuous at the given x-value:1.

2.

3.

7;52 xxxy

4;432

xxxy

2;24

26)(

2

x

xifx

xifxxf

End Behavior of a FunctionEven degree Positive leading coefficient

)(xf

x

)(xf

x

Even degree Negative leading coefficient

)(xf

x

)(xf

x

End Behavior of a FunctionOdd degree Positive leading coefficient

)(xf

x

)(xf

x

Odd degree Negative leading coefficient

)(xf

x

)(xf

x

ExampleDescribe the end behavior of the following

functions:35)( xxf

4245)( 23 xxxxg

ExampleGraph each function. Determine the

interval(s) on which the function increasing and the interval(s) on which the function is decreasing.

7)( 2 xxf

xxf 1)(

45)( 23 xxxxh

Monotonicity

A monotonic function is one that increases along rhe interval or decreases along the interval.