3-5 Continuity and End Behavior
Textbook Assignment: p. 166 # 12-14 (all), 16-27(all), 30
Notes/Examples:
Graphs of Discontinuous Functions – A function that cannot be drawn with a writing
device without a gap or picking up of the writing device. This is considered a
discontinuous function. There are either separate parts of the graph, or a hole(s) in the
graph. These functions are discontinuous at some point in their domain and show
"breaks" instead of a smooth, straight linear, or curved nonlinear line.
o Point Discontinuity – When there is a value in the domain for which the function
is undefined, but the pieces of the graph match up, then there is a point
discontinuity. Consider the piecewise function below:
 x 2 for x < 1
f ( x) = 
2 − x for x > 1
o Jump Discontinuity – A discontinuity where the function "jumps" from one
value to another. It is at this jump discontinuity that the graph indicates that it
stops at a given value of the domain and then begins again at a different range
value for the same value of the domain. Consider the piecewise function below:

x2
for x < 1
f ( x) = 
2
2 − ( x − 1) for x > 1
o Infinite Discontinuity – A type of discontinuity where the jump is infinite at a
given value of the domain. For example, when the graph of f ( x ) becomes
greater and greater as the graph approaches a given x-value, this is an infinite
discontinuity. Consider the graph below. You can see that as x gets closer to 4, the
graph of the range ( y ) will get closer and closer to infinity, an infinite jump:
Ex 1) Determine whether each graph is continuous at the given x -value. Give the type of
discontinuity at this value of the domain.
a) y = 3 x 2 + 7 ; x = 1
b) f ( x ) =
x2 − 4
; x = −2
x+2
Note: When you're graphing (or looking at a graph of) polynomials, it can help
to already have an idea of what basic polynomial shapes look like. One of the
aspects of this is "end behavior", and it's pretty easy.
End Behavior – describes what the range (y-values) of a function do as the domain
( x ) approaches positive and negative infinity ( f ( x ) → ? as x → ∞ and f ( x ) → ? as
x → −∞ )
Look at the end behavior of these graphs: Determine f ( x ) → ? as x → ∞ and
f ( x ) → ? as x → −∞ on each of the graphs below.
Notice that:
•
•
•
•
Even-degree polynomials are either "up" on both ends or "down" on both ends,
depending on whether the polynomial has a positive or negative leading coefficient.
Odd-degree polynomials have ends that head off in opposite directions. If they start
"down" and go "up", then they are positive polynomials; if they start "up" and go "down",
then they are negative polynomials.
All even-degree polynomials behave, on their ends, like quadratics.
All odd-degree polynomials behave, on their ends, like cubics.
Ex 2) Describe the end behavior of:
a)
f ( x ) = 5 x3
b) g ( x ) = −5 x3 + 4 x 2 − 2 x + 4
c) h ( x ) =
1 2
x −3
2
Determining whether a function is decreasing, constant, or increasing is always judged
by viewing a graph from left to right → . For example:
Ex 3) Determine the intervals on which the function is increasing, decreasing, or constant.
a)
f ( x ) = x2 − 7
b) f ( x ) = −
c)
f ( x ) = x3 + 3 x 2 − 9 x
d) f ( x ) =
e)
1
x
1
−4
x +1
f ( x ) = x2 − 9