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 y-values approach 11.

x	y = f(x)		x	y = f(x)	
1.9	9.659		2.1	12.461	
1.99	10.6806		2.01	11.1406	
1.999	10.986006		2.001	11.014006	

(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.

b.	f(x) = ; 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.

c.	
The function is defined at x = 0.
Using the first formula we find f(0) = 0

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	y = f(x)		x	y = f(x)	
-0.1	0.01		0.1	2.1	
-0.01	0.0001		0.01	2.01	
-0.001	0.000001		0.001	2.001	

	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)	Same-Day Shipping Rate
(dollars)	
	0-25 lbs.		$173	
	Each additional lb. in excess of 25 lbs. up to 70 lbs.		$1.25/lb.	
	Additional lbs. in excess of 70 lbs. (increments of 50 lbs.)		$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.

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 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.

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
x = -1 and x = . The function is
decreasing for x < -1. The function is
also decreasing for x > . When
-1 < x < , the function is increasing.