Warm up  Graph the function and its inverse:  Find for the relation. Then state whether is a function

Lesson 3-5 Continuity and End Behavior Objective: To determine whether a function is continuous or discontinuous To identify the end behavior of functions To determine whether a function is increasing or decreasing on an interval

Discontinuity Discontinuity – a break in the graph.  There are different types of discontinuity:  Infinite Discontinuity  Infinite Discontinuity :|f(x)| becomes greater & greater as the graph approaches a given value.

 Jump Discontinuity-  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-  Everywhere Discontinuous- impossible to graph in the real number system.  ex:

Which of the following does not display jump continuity? A D C B

Continuous  passes through all the points of the graph without a break.  Linear and quadratic functions are continuous.

Continuity Test – Continuity Test – must satisfy all 3 conditions  A 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).

Example  Determine whether each function is continuous at the given x-value.  1.  2.  3.

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 $ 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 up  Determine whether each function is continuous at the given x-value:  1.  2.  3.

End Behavior of a Function Even degree Positive leading coefficient Even degree Negative leading coefficient

End Behavior of a Function Odd degree Positive leading coefficient Odd degree Negative leading coefficient

Example  Describe the end behavior of the following functions:

Example  Graph each function. Determine the interval(s) on which the function increasing and the interval(s) on which the function is decreasing.

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