Page 44 - Calculus Demystified

CHAPTER 1
Basics
EXAMPLE 1.22 31
Let S be the set of all people and T be the set of all people. Let f be the
rule that assigns to each person his or her brother. Is f a function?
SOLUTION
In this case f is not a function. For many people have no brother (so the rule
makes no sense for them) and many people have several brothers (so the rule
is ambiguous for them).
EXAMPLE 1.23
Let S be the set of all people and T be the set of all strings of letters not
exceeding 1500 characters (including blank spaces). Let f be the rule that
assigns to each person his or her legal name. (Some people have rather
long names; according to the Guinness Book of World Records, the longest
has1063 letters.) Determine whether f : S → T isa function.

SOLUTION
This f is a function because every person has one and only one legal name.
Notice that several people may have the same name (such as “JackArmstrong”),
but that is allowed in the definition of function.

You Try It: Let f be the rule that assigns to each real number its cube root. Is this
a function?

In calculus, the set S (called the domain of the function) and the set T (called
the range of the function) will usually be sets of numbers; in fact they will often
consist of one or more intervals in R. The rule f will usually be given by one or
several formulas. Many times the domain and range will not be given explicitly.
These ideas will be illustrated in the examples below.

You Try It: Consider the rule that assigns to each real number its absolute value.
Is this a function? Why or why not? If it is a function, then what are its domain and
range?

1.8.1 EXAMPLES OF FUNCTIONS OF A REAL
VARIABLE

EXAMPLE 1.24
2
Let S = R, T = R, and let f(x) = x . Thisismathematical shorthand for
the rule “assign to each x ∈ S itssquare.” Determine whether f : R → R
isa function.
SOLUTION
We see that f is a function since it assigns to each element of S a unique
element of T —namely its square.