GRAPH OF A FUNCTION

GRAPH OF A FUNCTION

A function     f   establishes a set of ordered pairs             ًx; yق   of real numbers.           The plot of these pairs

ًx; f ًxقق in a coordinate system is the graph of f . The result can be thought of as a pictorial representa-

tion of the function.

BOUNDED FUNCTIONS

If there is a constant M such that fًxق @ M for all x in an interval (or other set of numbers), we say

that f   is bounded above in the interval (or the set) and call M an upper bound of the function.

If a constant m exists such that f ًxق A m for all x in an interval, we say that f ًxق is bounded below in

the interval and call m a lower bound.

CHAP. 3]

FUNCTIONS, LIMITS, AND CONTINUITY

41

If m @ f ًxق @ M in an interval, we call f ًxق bounded. Frequencly, when we wish to indicate that a

function is bounded, we shall write j f ًxقj < P.

EXAMPLES.            1.       f ًxق ¼ 3 ‏ x is bounded in                       1 @ x @ 1. An upper bound is 4 (or any number greater than 4).

A lower bound is 2 (or any number less than 2).

2.       f ًxق ¼ 1=x is not bounded in 0 < x < 4 since by choosing x su?ciently close to zero, f ًxق can be

made as large as we wish, so that there is no upper bound. However, a lower bound is given by

1

1