Page 50 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 3] FUNCTIONS, LIMITS, AND CONTINUITY 41

If m @ f ðxÞ @ M in an interval, we call f ðxÞ bounded. Frequencly, when we wish to indicate that a
function is bounded, we shall write j f ðxÞj < P.

EXAMPLES. 1. f ðxÞ¼ 3 þ x is bounded in 1 @ x @ 1. An upper bound is 4 (or any number greater than 4).
Alower bound is 2 (or any number less than 2).
2. f ðxÞ¼ 1=x is not bounded in 0 < x < 4 since by choosing x sufficiently close to zero, f ðxÞ can be
made as large as we wish, so that there is no upper bound. However, a lower bound is given by
1
1 (or any number less than ).
4 4
If f ðxÞ has an upper bound it has a least upper bound (l.u.b.); if it has a lower bound it has a greatest
lower bound (g.l.b.). (See Chapter 1 for these definitions.)

MONOTONIC FUNCTIONS
A function is called monotonic increasing in an interval if for any two points x 1 and x 2 in the interval
such that x 1 < x 2 , f ðx 1 Þ @ f ðx 2 Þ. If f ðx 1 Þ < f ðx 2 Þ the function is called strictly increasing.
Similarly if f ðx 1 Þ A f ðx 2 Þ whenever x 1 < x 2 , then f ðxÞ is monotonic decreasing; while if f ðx 1 Þ > f ðx 2 Þ,
it is strictly decreasing.

INVERSE FUNCTIONS. PRINCIPAL VALUES
Suppose y is the range variable of a function f with domain variable x. Furthermore, let the
1
correspondence between the domain and range values be one-to-one. Then a new function f , called
the inverse function of f , can be created by interchanging the domain and range of f . This information is
contained in the form x ¼ f 1 ðyÞ.
As you work with the inverse function, it often is convenient to rename the domain variable as x and
use y to symbolize the images, then the notation is y ¼ f 1 ðxÞ. In particular, this allows graphical
expression of the inverse function with its domain on the horizontal axis.
Note: f 1 does not mean f to the negative one power. When used with functions the notation f 1
always designates the inverse function to f .
If the domain and range elements of f are not in one-to-one correspondence (this would mean that
distinct domain elements have the same image), then a collection of one-to-one functions may be created.
Each of them is called a branch. Itis often convenient to choose one of these branches, called the
principal branch,and denote it as the inverse function, f 1 . The range values of f that compose the
principal branch, and hence the domain of f 1 , are called the principal values. (As will be seen in the
section of elementary functions, it is common practice to specify these principal values for that class of
functions.)

EXAMPLE. Suppose f is generated by y ¼ sin x and the domain is 1 @ x @ 1. Then there are an infinite
number of domain values that have the same image. (A finite portion of the graph is illustrated below in Fig. 3-2(a.)
In Fig. 3-2(b)the graph is rotated about a line at 458 so that the x-axis rotates into the y-axis. Then the variables are
interchanged so that the x-axis is once again the horizontal one. We see that the image of an x value is not unique.
Therefore, a set of principal values must be chosen to establish an inverse function. A choice of a branch is

accomplished by restricting the domain of the starting function, sin x. For example, choose @ x @ .
2 2
Then there is a one-to-one correspondence between the elements of this domain and the images in 1 @ x @ 1.
Thus, f 1 may be defined with this interval as its domain. This idea is illustrated in Fig. 3-2(c) and Fig. 3-2(d).
With the domain of f 1 represented on the horizontal axis and by the variable x,wewrite y ¼ sin 1 x, 1 @ x @ 1.

1
If x ¼ ,then the corresponding range value is y ¼ .
2 1 6
Note:Inalgebra, b 1 means and the fact that bb 1 produces the identity element 1 is simply a rule of algebra
b
generalized from arithmetic. Use of a similar exponential notation for inverse functions is justified in that corre-
sponding algebraic characteristics are displayed by f 1 ½ f ðxފ ¼ x and f ½ f 1 ðxފ ¼ x.