Functions Worksheet With Answer Key Page 16
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4684
Chapter Two FUNCTIONS
(c) What is the domain of s(r)? Based on your graph,
39. The surface area of a cylindrical aluminum can is a mea-
what, approximately, is the range of s(r)?
sure of how much aluminum the can requires. If the can
has radius r and height h, its surface area A and its vol-
(d) The manufacturers wish to use the smallest amount
2
ume V are given by the equations:
of aluminum (in cm
) necessary to make a 12-oz
cola can. Use your answer in (c) to find the mini-
2
2
A = 2πr
+ 2πrh and V = πr
h.
mum amount of aluminum needed. State the values
of r and h that minimize the amount of aluminum
used.
3
(a) The volume, V , of a 12-oz cola can is 355 cm
.
(e) The radius of a real 12-oz cola can is about 3.25 cm.
A cola can is approximately cylindrical. Express its
Show that real cola cans use more aluminum than
surface area A as a function of its radius r, where r
necessary to hold 12 oz of cola. Why do you think
is measured in centimeters. [Hint: First solve for h
real cola cans are made in this way?
in terms of r.]
(b) Graph A = s(r), the surface area of a cola can
3
, for 0 ≤ r ≤ 10.
whose volume is 355 cm
2.3
PIECEWISE-DEFINED FUNCTIONS
A function may employ different formulas on different parts of its domain. Such a function is said to
be piecewise defined. For example, the function graphed in Figure 2.15 has the following formulas:
2
2
y = x
for x ≤ 2
x
for x ≤ 2
or more compactly y =
y = 6
x for x > 2
6
x for x > 2.
y
✲
✛
2
y = 6
x
y = x
x
2
:
Figure 2.15
Piecewise defined function
Example 1
Graph the function y = g(x) given by the following formulas:
g(x) = x + 1
x ≤ 2
g(x) = 1
x > 2.
for
and
for
Using bracket notation, this function is written:
x + 1
for x ≤ 2
g(x) =
1
for x > 2.
Solution
For x ≤ 2, graph the line y = x + 1. The solid dot at the point (2, 3) shows that it is included in
the graph. For x > 2, graph the horizontal line y = 1. See Figure 2.16. The open circle at the point
(2, 1) shows that it is not included in the graph. (Note that g(2) = 3, and g(2) cannot have more
than one value.)
y
4
g(x)
3
2
1
3 2 1
x
1 2 3 4
1
2
:
Figure 2.16
Graph of the piecewise defined function g
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