Bridge to Abstract Mathematics: Mathematical Proof and Structures

8.1 FUNCTIONS AND MAPPINGS 259

Figure 8.3 Graph of a function that is its own inverse. Each point
(x, y) on the graph of the function y = llx has its corresponding
point (y, x) also lying on the curve.

f, it must have happened that no two distinct ordered pairs had the same
second element. The next theorem formalizes this relationship.

THEOREM 1

Let f be a function. Then the inverse relation f-' is a function if and only if

f is one to one.

Proof Assume that f - ' is a function. To prove f is one to one, choose

xl, X, E dom f and assume f(xl) = f(x,); we must prove x, = x,. Let-

ting y = f (x ,) = f (x,), we have that the ordered pairs (y, x,) and (y, x,)

are both in f -' (why?). Since f -' is a function, we conclude x, = x,,

as desired. The converse is left as an exercise [Exercise 5(a)]. 0

Many authors express this theorem as "iff is a function, then f - exists

(or f has an inverse) if and only if f is one to one." Since f -' always

exists as a relation, such a formulation is, strictly speaking, inaccurate.

What is meant by this, of course, is that f -l exists as a function precisely

when f is one to one. We say that f is invertible in this case.

A second general method of creating a function from a given function

is suggested by the difficulties that are faced in defining the inverse trigo-

nometric functions. As you know, functions such as sin, cos, and tan, are