Download Piecewise and Absolute Value Examples

Piecewise and Absolute Value Functions
Example 1
3 if x -1
Graph f ( x ) = 4 - x if - 1 x 4 .
3 x - 10 if x 4
First, graph the constant function f ( x) 3 for x 1 . This graph is part of a horizontal line. Because the
point at (-1, 3) is included in the graph, draw a closed circle at that point.
Second, graph the function f ( x) 4 x for
1 x 4 . Because x 1 is not included
in this part of the domain, draw an open circle
at (-1, 5). x 4 is included in the domain, so
draw a closed circle at (4, 0) since for
f ( x) 4 x, f (4) 0 .
Third, graph the line y 3x 10 for x 4 .
Draw an open circle at (4, 2) since for
f ( x) 3x 10, f (4) 2 .
Example 2
Graph f ( x ) = -2 x .
Make a table of values. The domain values will be intervals for which the greatest integer function will be
evaluated.
-3
-2
-1
0
1
2
3
4
x
x < -2
x < -1
x<0
x<1
x<2
x<3
x<4
x<5
f(x)
6
4
2
0
-2
-4
-6
-8
Notice that the domain for this greatest integer function is all real numbers and the range is even integers.
Example 3
POSTAGE The amount of postage a person has to
pay in order to mail a parcel via first class mail
depends upon the weight of the parcel. The table
shows the cost of postage for each of the weight
classes up to 6 ounces.
a. Graph the cost postage for the different parcel
weights.
b. What is the cost of postage for a parcel that
weighs 4.2 ounces?
First-Class Mail
Postage Rates
Weight
Cost of
(ounces)
Postage ($)
0.37
0<x 1
0.60
1<x 2
0.83
2<x 3
1.06
3<x 4
1.29
4<x 5
1.52
5<x 6
b. 4.2 ounces falls in the interval 4 ounces to
5 ounces. Thus, the cost of a parcel
weighing 4.2 ounces is $1.29.
a.
Example 4
1
Graph f(x) = |x| + 1.
2
Use a table of values to determine points on the graph.
x
-4
-2
0
2
4
6
1
|x| + 1
2
3
2
1
2
3
4
(x, f(x))
(-4, 3)
(-2, 2)
(0, 1)
(2, 2)
(4, 3)
(6, 4)
Example 5
Identify the type of function that models each situation. Then write a function
for the situation.
a. The acceptable temperature range to store
refrigerated foods is between 34 F and 40 F.
Therefore, the temperature of a refrigerator may
be within a 3-degree range of 37 F. Outside of
this range, and food either begins to freeze or
spoil.
The situation can be represented with an absolute
value function. Let t represent the temperature and
d(t) represent the discrepancy. Then d (t ) 37 t .
b. A credit card company is running a rewards
program in which its card members earn one
reward point for every dollar they spend using
their credit card. Points are only awarded in
whole number increments for each whole dollar
spent. Therefore, if a person spends $1.00 using
her card, she earns 1 reward point, but if she
spends $1.70, she will still only earn 1 reward
point.
This can be described by a greatest integer function.
Let d represent the number of dollars spent and
p(d ) represent the points earned.
p d
d if d
d if d
d
d