Continuity and End Behavior

Section 3-5

Learning Target

• I can determine whether a function is continuous or discontinuous

• I can identify the end behavior of functions

• I can determine whether a function is increasing or decreasing on an interval

Continuous Function-the graph of this function is smooth or continuous curves. (Linear and Quadratic Function)

Discontinuous function- the graph of this function cannot be trace without lifting your pencil (step wise and absolute value).

The chart below shows the different types of discontinuous functions.

2

1y

x

     

Function Name Graph Characteristics

    

 Infinite Discontinuity

 

The graph becomes greater and greater as it approaches a given x-value.

2 1 0( )

0

x if xf x

x if x

2 1( )

1

xf x

x

      

    

     JumpDiscontinuity

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

When there is a value in the domain for which the function is undefined, but the pieces of the graph match up. There is a hole in the graph.

Function Name Graph Characteristics

A function is continuous at x = c if it satisfies the following conditions:

1. the function is defined at c; or f(c) exists2. the function approaches the same y-

value on the left and right sides of x = c3. the y-value that the function approaches

from each side is f(c).

Continuity Test

Example: Determine whether the function is continuous at the given x-value.

2 4( ) ; 2

2

xf x x

x

Determine whether each function is continuous at the given x-value.

a. f(x) = x3 + 2x - 1; x = 2

b. f(x) = x2 + 5x + 6x + 3 ; x = -3

c.

2 if 0( ) = ; 0

2 if 0x x

f x xx x

Another tool that is used for analyzing functions is end behavior. End behavior is just a way to describe what happens to the graph at the ends. In the table below “leading coefficient” refers to the coefficient in the term of the polynomial with the highest power of x. For instance, 3 is the leading coefficient in the polynomial 4 3( ) 3 2 4 7f x x x x .

Polynomial Properties End Behavior Example

Leading coefficient is positive; highest power

is even

As x gets very large, y gets very large; as x

gets very small, y gets very large.

2( ) 3f x x ( )f x as x ( )f x as x

Leading coefficient is negative; highest power

is even

As x gets very large, y gets very small; as x

gets very small, y gets very small.

2( ) 3f x x ( )f x as x

( )f x as x

Leading coefficient is positive; highest power is

odd

As x gets very large, y gets very large; as x gets

very small, y gets very small.

3( ) 2f x x ( )f x as x

( )f x as x

Leading coefficient is negative; highest power

is odd

As x gets very large, y gets very small; as x gets

very small, y gets very large.

3( ) 2f x x ( )f x as x ( )f x as x

Polynomial Properties End Behavior Example

Another characteristic of functions used for analysis is the monotonicity of the function. This means that on an interval, the function is increasing or decreasing on that particular interval. Whether a graph is increasing or decreasing is always judged by viewing a graph from left to right.

Function Characteristics Increasing A function f is increasing on an interval

if and only if for every a and b contained in the interval, f(a)<f(b)

whenever a<b Decreasing A function f is increasing on an interval

if and only if for every a and b contained in the interval, f(a)>f(b)

whenever a<b Constant A function f is increasing on an interval

if and only if for every a and b contained in the interval, f(a)=f(b)

whenever a<b

Example: Describe the end behavior of the function. Graph the function and determine the intervals on which the function is increasing and the intervals on which the function is decreasing. 3 2( ) 5 4f x x x x The leading coefficient is positive and the highest power is odd. Therefore, as

,x y and as ,x y .

Use your graphing calculator to find the relative max and relative min to find your boundaries for the increasing and decreasing intervals.

Describe the end behavior of f(x) = 3x2 + 4x + 1 and g(x) = -2x2 + 2x. Use your calculator to create a table of function values so you can investigate the end behavior of the y-values.

f(x) = 3x2 + 4x + 1 g(x) = -2x2 + 2x x f(x) x g(x)

-10,000 299,960,001 -10,000 -200,020,000 -1000 2,996,001 -1000 -2,002,000 -100 29,601 -100 -20,200 -10 261 -10 -220 0 1 0 0 10 341 10 -180

100 30,401 100 -19,800 1000 3,004,001 1000 -1,998,000

10,000 300,040,001 10,000 -199,980,000

Notice that both polynomial functions have y-values that become very large in absolute value as x gets very large in absolute value. The end behavior of f(x) can be summarized by stating that as x

, f(x) and as x -, f(x) . The end behavior of g(x) can be summarized by stating

that as x , g(x) - and as x - , g(x) -. You may wish to graph these functions on a

graphing calculator to verify this summary.

Graph each function. Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. a. f(x) = (x + 1)2 - 4

The graph of this function is obtained by transforming the parent graph p(x) = x2. The parent graph has been translated 1 unit to the left, and translated down 4 units. The function is decreasing for x < -1 and increasing for x > -1. At x = -1, there is a critical point.

b. f(x) = |x - 3| + 2 The graph of this function is obtained by transforming the parent graph p(x) = |x|. The parent graph has been reflected about the x-axis, translated 3 units right, and translated up two units. This function is increasing for x < 3 and decreasing for x > 3. There is a critical point when x = 3.

c. f(x) = -x3 - x2 + x + 2 This function has more than one critical point. It changes direction at

x = -1 and x = 13. The function is

decreasing for x < -1. The function is

also decreasing for x > 13. When

-1 < x < 13, the function is increasing.

Helpful Websites

Discontinuity:

http://www.sparknotes.com/math/precalc/continuityandlimits/problems3.rhtml

http://math.usask.ca/~maclean/101/Limits/Printables/BW/Continuity.pdf

End behavior: http://www.purplemath.com/modules/polyends.htm

3-5 Self Check Quiz: http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-860861-9&chapter=3&lesson=5&quizType=1&headerFile=4&state=