Page 45 - Calculus Demystified

Basics
CHAPTER 1
32
Math Note: Notice that, in the definition of function, there is some imprecision
in the definition of T . For instance, in Example 1.24, we could have let T =[0, ∞)
or T = (−6, ∞) with no significant change in the function. In the example of the
“name” function (Example 1.23), we could have let T be all strings of letters not
exceeding 5000 characters in length. Or we could have made it all strings without
regard to length. Likewise, in any of the examples we could make the set S smaller
and the function would still make sense.
It is frequently convenient not to describe S and T explicitly.

EXAMPLE 1.25
√
2
Let f(x) =+ 1 − x . Determine a domain and range for f which make
f a function.
SOLUTION
Notice that f makes sense for x ∈[−1, 1] (we may not take the square root
of a negative number, so we cannot allow x> 1or x< −1). If we understand
f to have domain [−1, 1] and range R, then f :[−1, 1]→ R is a function.
Math Note: When a function is given by a formula, as in Example 1.25, with no
statement about the domain, then the domain is understood to be the set of all x for
which the formula makes sense.

You Try It: Let
x
g(x) = .
2
x + 4x + 3
What are the domain and range of this function?
EXAMPLE 1.26
Let

−3 if x ≤ 1
f(x) = 2
2x if x> 1
Determine whether f isa function.
SOLUTION
Notice that f unambiguously assigns to each real number another real num-
ber. The rule is given in two pieces, but it is still a valid rule. Therefore it is
a function with domain equal to R and range equal to R. It is also perfectly
correct to take the range to be (−4, ∞), for example, since f only takes values
in this set.

Math Note: One point that you should learn from this example is that a function
may be specified by different formulas on different parts of the domain.