384
Relations, Functions, and Matrices
PRaCtiCe 23 Which of the following formulas are functions from the domain to the codomain indicated? For those that are not, why not?
a. f : S S T where S = T = 51, 2, 36, f = 5(1, 1), (2, 3), (3, 1), (2, 1)6
b. g: Z S N where g is defined by g(x) = 0 x 0 (the absolute value of x)
c. h: N S N where h is defined by h(x) = x − 4
d. f: S S T where S is the set of all people in your hometown, T is the set of all automobiles, and f
associates with each person the automobile that person owns
e. g: S S T where S = 52013, 2014, 2015, 20166, T = 5$20,000, $30,000, $40,000, $50,000,
$60,0006, and g is defined by the graph in Figure 5.16.
f. h: S S T where S is the set of all quadratic polynomials in x with integer coefficients, T = Z, and
h is defined by h(ax
2
+ bx + c) = b + c
g. f: R S R where f is defined by f (x) = 4x − 1
h. g: N S N where g is defined by
g (x) = e
x + 3 if x ≥ 5
x     if x ≤ 5
2013
2014
2015
Year
2016
60
50
40
30
Profit in thousands of dollars
20
10
0
Figure 5.16 Profits of the American Earthworm Corp.
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PRaCtiCe 24 Let f : Z S Z be defined by f (x) = x
2
.
a. What is the image of −4?
b. What are the preimages of 9?
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example 27
When we studied recursive definitions in Section 3.1, we talked about sequences,
where a sequence S was written as
S(1), S(2), S(3), …
Changing the notation to
f (1), f (2), f (3), …
we see that a sequence is nothing but a list of functional values for a function f
whose domain is the positive integers, and this is how a sequence is often defined.