Download 3-5 Continuity and End Behavior

Advanced Mathematical Concepts
Chapter 3
Lesson 3-5
Example 1
Determine whether each function is continuous at the given x-value.
a. f(x) = x3 + 2x - 1; x = 2
Check the three conditions in the continuity test.
(1) The function is defined at x = 2. In particular, f(2) = 11.
(2) The first table below suggests that when x is less than 2 and x approaches 2, the y-values
approach 11. The second table suggests that when x is greater than 2 and x approaches 2, the yvalues approach 11.
x
1.9
1.99
1.999
y = f(x)
9.659
10.6806
10.986006
x
2.1
2.01
2.001
(3) Since the y-values approach 11 as x approaches
2 from both sides and f(2) = 11, the function is
continuous at x = 2. This can be confirmed by
examining the graph.
x2 + 5x + 6
; x = -3
x+3
Start with the first condition in the continuity test.
The function is not defined at x = 3 because
substituting -3 for x results in a denominator of
zero. So the function is discontinuous at x = -3.
This function has point discontinuity at x = -3.
b. f(x) =
y = f(x)
12.461
11.1406
11.014006
Advanced Mathematical Concepts
c.
Chapter 3
if x  0
;x  0
x

2
if
x0

The function is defined at x = 0.
Using the first formula we find f(0) = 0
 x2
f ( x) = 
The first table suggests that f(x) approaches 0 as x approaches 0 from the left. We can see from the
second table that f(x) seems to approach 2 as x approaches 0 from the right.
x
-0.1
-0.01
-0.001
y = f(x)
0.01
0.0001
0.000001
y = f(x)
2.1
2.01
2.001
x
0.1
0.01
0.001
Since the f(x) values approaches different values
as x approaches 0 from both sides, the function is
discontinuous at x = 0. The graph has jump
discontinuity at x = 0.
Example 2
SHIPPING Zofia needs to ship a package for
same day delivery. The charges for same-day
delivery are shown in the table. The graph
summarizes the cost of same-day delivery
service.
Package Weight
(pounds)
0-25 lbs.
Each additional lb. in
excess of 25 lbs. up to
70 lbs.
Additional lbs. in
excess of 70 lbs.
(increments of 50 lbs.)
Same-Day
Shipping Rate
(dollars)
$173
$1.25/lb.
$85/50lbs.
a. Use the continuity test to show that the piecewise function is discontinuous.
b. Explain why a continuous function would not be appropriate to model same-day delivery rates.
Advanced Mathematical Concepts
Chapter 3
a. The graph of same-day shipping function is discontinuous at each integral value of w in its domain
because the function does not approach the same value from the left and the right. For example, as w
approaches 70 from the left, C(w) approaches 229.25, but as w approaches 70 from the right, C(w)
approaches 314.25.
b. A continuous function would have to achieve all real y-values (greater than or equal to 173.) This
would be an inappropriate model for this situation since the weight of a package is rounded to the
nearest ounce and shipping costs are rounded to the nearest cent.
Example 3
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 yvalues.
f(x) = 3x2 + 4x + 1
f(x)
x
-10,000
299,960,001
-1000
2,996,001
-100
29,601
-10
261
0
1
10
341
100
30,401
1000
3,004,001
10,000
300,040,001
g(x) = -2x2 + 2x
g(x)
x
-10,000
-200,020,000
-1000
-2,002,000
-100
-20,200
-10
-220
0
0
10
-180
100
-19,800
1000
-1,998,000
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.
Advanced Mathematical Concepts
Chapter 3
Example 4
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
1
x = -1 and x = 3. The function is
decreasing for x < -1. The function is
1
also decreasing for x > 3. When
1
-1 < x < , the function is increasing.
3